Blaise Pascal (1623–1662) - mathtoday.files.wordpress.com · កម វិធីគណនាប េផ្ស្គណិៗក កណវិវិទមយោត ោភាស. By

Blaise Pascal (1623–1662)

កមវធគណនាប េផសកៗក គណណវវទមយោត ោភ ាស

By Phalkun Lim Page 1

មណកកេ

១.វយគាបផ ១.១.ហ ក តែរ

១.២.ចមា ក ១.៣.បនស

២.េាាកសកណបសក ២.១.ផបត នពន

២.២.ផបត នពនសដបក ខក ២.៣.ផបត ធែណមា

២.៤.ផបត ដមនត ផផណែណរ នពនននរ ធែណមា ២.៥.ផបត ចសនតនពនេផរង ២.៦.ផផណ ចសនតនពនេផរង ៣.ណវn ៃបសកណទមកាមបនកណណ ប ៣.១.តទn ៃន ដមនែតបបនសេណន 1n nu au b+ = +

៣.២.តទn ៃន ដមនែតបបនសេណន 1n

nn

au bucu d+

+=

+

៣.៣.តទn ៃន ដមនែតបបនសេណន 21n n nu au bu c+ = + +

៣.៤.តទn ៃន ដមនែតបបនសេណន 3 21n n n nu au bu cu d+ = + + +

៣.៥.តទn ៃន ដមនែតបបនសេណន2

1 ' 'n n

nn

au bu cua u b+

+ +=

+

៣.៦.តទn ៃន ដមនែតបបនសេណន2

1 1 11 2

2 2 2

n nn

n n

a u b u cua u b u c+

+ +=

+ +

៣.៧.តទn ៃន ដមនែតបបនសេណន 2 1n n nu au bu+ += + ៣.៨.តទn ៃន ដមនែតបបនសេណន 2 1n n nu au bu c+ += + +

៣.៩.តទn ៃន ដមនែតបបនសេណន 21n n n nu u u au b+ = + + +

៣.១០.តទn ៃន ដមនែតបបនសេណន 1n nu au b+ = +

៣.១១.តទn ៃន ដមនែតបបនសេណន 21 2n nu u+ = −

៣.១២.តទn ៃន ដមនែតបបនសេណន 31 3n n nu u u+ = −

៤.អបកគមបណណមណ

៤.១.េដយស កលsina ផណនៃបាលcos ,tan ,cota a a ៤.២.េដយស កcosa ផណនៃបាលsin ,tan ,cota a a



៤.៣.េដយស កលtan a ផណនៃបាលsin ,cos ,cota a a ៤.៤.េដយស កលcot a ផណនៃបាលsin ,cos ,tana a a

៤.៥.េដយស កល tan2at = ផណនៃបាលsin ,cos ,tan ,cota a a a

៤.៦.េដយស កលsina ផណនលsin3a

៤.៧.េដយស កលcosa ផណនលcos3a ៤.៨.េដយស កលtan a ផណនលtan3a

៤.៩.េដយស កលcot a ផណនលcot3a ៥.ណណបកស

៥.១.េដយស ការជសរបៃនាេណABC ផណនៃផផារែបកណ ៥.២.េដយស ការជសរបៃនាេណABC ផណនែរ កសែររកែេាៃនាេណ

៥.៣.េដយស ការជសរបៃនាេណABC ផណនែរ កសែររកែររៃនាេណល ៥.៤.េដយស ការជសរបៃនាេណABC ផណនលsin , sin , sinA B C ៥.៥.េដយស ការជសរបៃនាេណABC ផណនលcos , cos , cosA B C ៥.៦.េដយស ការជសរបៃនាេណABC ផណនល tan , tan , tanA B C ៥.៧. េដយស ការជសរបៃនាេណABC ផណនលcot , cot , cotA B C ៥.៨.េដយស ការជសរបៃនាេណABC ផណនែរ កសែររកែររបសៃនាេណល៥.៩.េដយស ការជសរបៃនាេណABC ផណនែរ កេបដដនជសរបៃនាេណ ៥.១០.េដយស ការជសរបៃនាេណABC ផណនែរ កបខកជសរបៃនាេណ ៥.១១.េដយស ការជសរបៃនាេណABC ផណនាបែរងរង កពបសររជសរបៃនាេណ ៥.១២.េដយស ការជសរបៃនាេណABC ផណន , , , ,OG OI OH IH IG ននរGH ដOផផនែររកែររ I ផផនែររកែេាGទាបសទបទនកននរH ងែ តរកៃនាេណ

៥.១៣.េដយស ការជសរបៃនាេណABC ផណនលsin sin sinA B B+ + ៥.១៤.េដយស ការជសរបៃនាេណABC ផណនcos cos cosA B C+ +

៥.១៥.េដយស ការជសរបៃនាេណABC ផណនល tan tan tanA B C+ + ៥.១៦.េដយស ការជសរបៃនាេណABC ផណនcot cot cotA B C+ +

៥.១៧.េដយស ការជសរបៃនាេណABC ផណនលsin ,sin ,sin2 2 2A B C

៥.១៨.េដយស ការជសរបៃនាេណABC ផណនលcos ,cos ,cos2 2 2A B C

៥.១៩.េដយស ការជសរបៃនាេណABC ផណនល tan ,tan ,tan2 2 2A B C

៥.២០.េដយស ការជសរបៃនាេណABC ផណនលcot ,cot ,cot2 2 2A B C



៥.២១.េដយស ការជសរបៃនាេណABC ផណនល tan tan tan2 2 2A B C+ +

៥.២២.េដយស ការជសរបៃនាេណABC ផណនលcot cot cot2 2 2A B C

+ +

៥.២៣.េដយស ការជសរបៃនាេណABC ផណនែរ កបសBrocard ៦.ចណកណត

៦.១.ៃផផារចេណេណ រេដយស កាបែរារននរផបតបសទ កសពតពែ ៦.២.ៃផផារចេណេណ រេដយស កាបែរារននរែរ កបសទ កសពតពែល

៦.៣.ៃផផារចេណេណ រេដយស កាបែរងរង កាទរជសរពែននរបសែណរងរង កជសរពែ ៧.ចណកណត ក ចកកៗក កស

៧.១.ៃផផារចេណេណ រែររែររកេដយស កាបែរារជសរបតន ៧.២.ផផណងរង កាទរជសរពែៃនចេណេណ រែររែររកេដយស កាបែរារជសរបតន

៧.៣.ផេធៀបងរង កាទរជសរពែៃនចេណេណ រែររែររកេដយស កាបែរារជសរបតន ៧.៤.ាបែរងរង កាទរជសរពែៃនចេណេណ រែររែររកេដយស កាបែរារជសរបតន ៨.កអកដបកៗក ាបស ៨.១.តងែេដេនចសណចក ៃនងរង កបត ៨.២.ចមទ ែណរពែចសណចររបារក ៨.៣.តងែេដេនៃនចសណចចររងរង កបត ៨.៤.តងែេដេនៃនចសណចចេាងរង កបត ៨.៥.តងែេដេនទាបសទបទនកៃនាេណបត ៨.៦.ាបែរារៃនាេណបតេដយស កតងែេដេនសពត ៨.៧.តងែេដេនផផនែររកែររាេណបត ៨.៨.ៃផផារាេណបតយស កតងែេដេនសពត ៨.៩.ែរ កសែររកែររាេណបតេដយស កតងែេដេនសពត ៨.១០.ែរ កសែររកែេាាេណបតេដយស កតងែេដេនសពត

៨.១១.តងែេដេនៃនផផនែររកែេាាេណបត ៨.១២.តងែេដេនងែ តរកៃនាេណបត

៨.១៣.តងែេដេនៃន , , ,G O H I ៨.១៤.ាបែរល , , ,OG OI OH IH ននរ IG

៨.១៥.ាបែរេបដដនជសរបៃនាេណេដយស កតងែេដេនសពត ៨.១៦.ាបែរបខកជសរបៃនាេណេដយស កតងែេដេនសពត

៨.១៧.ាបែរងរង កពបសររជសរបៃនាេណេដយស កតងែេដេនសពត ៨.១៨.ចមទ ពសពតបតេេចសណចចររៃនារជបែបកាេណ



៨.១៩.ចមទ ពសពតបតេេចសណចចេាៃនារជបែបកាេណ ៨.២០.ាបែរសែររកែររបសៃនាេណ

៨.២១.ចមទ ពចសណចបតេេបនផ កបតររបារក ៨.២២.ចមទ ែណរបនផ កពែាប រ

៨.២៣.តងែេដេនៃនចសណចាបពែណរបនផ កពែ ៨.២៤.តងែេដេនសពតាេណនបពែបនផ កបាបព រ ៨.២៥.បែែររកែេាាេណដនបពបនផ កបាបព រ

៨.២៦.តងែេដេនផផនែររកេដយស កសេកបណេេនរបនផ កពែដាបព រ ៩.ធកណមណកៗក ាល

៩.១.តងែេដេនចសណចក ៃនងរង កបត ៩.២.ចមទ ែណរពែចសណចររសក

៩.៣.តងែេដេនទាបសទបទនកាេណររសក ៩.៤.តងែេដេនចសណចចររៃនងរង កបតររសក ៩.៥.តងែេដេនចសណចចេាៃនងរង កបតររសក ៩.៦.ផផណយង ែណរពែែ នចទចែររសក ៩.៧.តនតៃនបសែណរពែែ នចទចែ ៩.៨.ាបែរារាេណររសក ៩.៩.តងែេដេនផផនែររកែររាេណររសក ៩.១០.ៃផផារននរសែររកែេាាេណររសក ៩.១១.ៃផផារសែររកែររាេណររសក ៩.១២.ផផណៃនពែែ នចទចែររសក ៩.១៣.ផផណចាប ៩.១៤.ផផណៃនបែ នចទចែររសក ៩.១៥.ផផណចាបៃនបតនែ នចទចររសក

៩.១៦.ៃផផារាេណររសក ៩.១៧.មាចបខេាង

៩.១៨.ចមទ ពចសណចបតេេបារកររសក ៩.១៩.ចមទ ពចសណចបតេេបនផ កររសក ៩.២០.ចមទ ែណរបារកពែាប រ ៩.២១.ចមទ ែណរបនផ កពែររសក ៩.២២.តងែេដេនចសណចាបពែណរបនផ កននរបារក ៩.២៣.តងែេដេនចសណចេរៃនចសេករែបកចសណចបតេេេបារកររសក



៩.២៤.តងែេដេនចសណចចា រ េធៀបននរបារកបតររសក ៩.២៥.បែបណននរបារកបតររសកននរតងែេដេនៃនចសណចបណ

៩.២៦.តងែេដេនៃនផផនែបកបណននរបារការកចសណចបត ៩.២៧.តងែេដេនៃនផផនននរាបែរសែបកែេាចបបត

៩.២៨.តងែេដេនងែ តរកននរផផនែររកែេាាេណររសក ៩.២៩.តងែេដេនចសណចាបពែណរបារកបល

www.mathtoday.wordpress.com

http://www.mathtoday.wordpress.com/



១.វយគាបផ

១.១.ហ កសណតែរា

ផលគណៃន n ចននគតវជមនន បងនលលណតេដ ! 1 2 3 4 ....n n= × × × × ×

ហn ហ លតបងែរល ល

លមវធគណន ! 1 2 3 ....n n= × × × × var n,i:integer; f:real; begin write('input n= '); read(n); f:=1; for i:=1 to n do f:=f*i; writeln('n!=',f:12:12);readln; End.

១.២.ចមា ស

តមមn ធតស គៗ ាដលមយ kធតងនលលលទផលគតលេហចមថ ងនលលណត

េដ ( ) ( )( ) ( )!, 1 2 ..... 1( )!

nP n k n k n k n nn k

= = − + − + −−

ល

លមវធគណន ( ) ( )!,

!nP n k

n k=

−

var n,i,j,k:integer; f,p,g:real; begin write('input n= '); read(n); write('input k= '); read(k); begin f:=1;g:=1; for i:=1 to n do f:=f*i; end; for j:=1 to (n-k) do g:=g*j; p:=f/g; writeln('P(n,k)=',p:12:12);readln; End.



១.៣.ាបផ

តមមn ធតស គៗ ាដលមយ kធតងនលលលទផលមនគតលេហចមថ ងនលលណត

េដ ( ) ( )!,

! !kn

n nC n k Cn k kk

= = = −

ល

លមវធគណន ( ) ( )!,

! !nC n k

k n k=

−

var n,i,j,k,l:integer; f,p,q,g,c:real; begin write('input n= '); read(n); write('input k= '); read(k); begin f:=1;g:=1; for i:=1 to n do f:=f*i; end; begin for j:=1 to (n-k) do g:=g*j; p:=f/g; end; begin q:=1; for l:=1 to k do q:=q*l; end; c:=p/q; writeln('C(n,k)=',c:12:12);readln; End.





២.េាាតកកណប ែ ២.១.េាាតកកណបពប

ក.តទn ៃនសតនពន

( )1 1nu u n d= + − ងនល 1u ជតន ប ន d ជផលែមៃន តល

លមវធគណនត :nu var n,i:integer; u1,un,d:real; begin write('u1= ');read(u1); write('d= ');read(d); write('n= ');read(n); un:=u1; for i:=1 to n-1 do un:=un+d; writeln('un=' ,un:8:12);readln; End.

ខ.របមនផលបកn តដបងៃនសតនពន

( ) ( ) 111 2 3

1

2 ( 1)...

2 2n

nn i n

i

n u n dn u uS u u u u u

=

+ −+ = = + + + + = =∑

លមវធគណនផល បល :nS var n,i:integer; u1,un,d,sn:real; begin write('u1= ');read(u1); write('d= ');read(d); write('n= ');read(n); sn:=u1; for i:=1 to n-1 do sn:=sn+u1+i*d; writeln('sn=' ,sn:8:12);readln; End.

២.២.េាាតកកណបពបាដាសខស

លមវធគណនផល បល 2 2 2 21 2 3 ...nS n= + + + + var n,i:integer; sn:real; begin write('n= ');read(n); sn:=0 ; for i:=1 to n do sn:=sn+i*i; writeln('sn=' ,sn:8:12);readln; End.



លមវធគណនផល បល 3 3 3 31 2 3 ...nS n= + + + + var n,i:integer; sn:real; begin write('n= ');read(n); sn:=0 ; for i:=1 to n do sn:=sn+i*i*i; writeln('sn=' ,sn:8:12);readln; End.

លមវធគណនផល បល 4 4 4 41 2 3 ...nS n= + + + + var n,i:integer; sn:real; begin write('n= ');read(n); sn:=0 ; for i:=1 to n do sn:=sn+i*i*i*i; writeln('sn=' ,sn:8:12);readln; End.

លមវធគណនផល បល 1 2 3 ...p p p pnS n= + + + +

var n,i:integer; sn,p:real; begin write('n= ');read(n); write('p= ');read(p); sn:=0 ; for i:=1 to n do sn:=sn+exp(p*ln(i)); writeln('sn=' ,sn:8:12);readln; End.

២.៣.េាាតកកណធែណមាណ

ក.តទn ៃនសតធរណមត

11

nnu u q −= × ងនល 1 :u ជតន ប ន q ជផលធមែមល

លមវធគណនតល :n var n,i:integer; u1,un,q:real; begin write('u1= ');read(u1); write('q= ');read(q); write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=un*q; writeln('un=' ,un:8:12);readln; End.



ខ.ផលបក n តដបងៃនសតធរណមត

( ) 1 2 3 11

1...1

nn

n i ni

qS u u u u u uq=

−= = + + + + = ×

−∑ ល

លមវធគណនផល បល តធែណមត :nS var n,i:integer; u1,sn,q:real; begin write('u1= ');read(u1); write('q= ');read(q); write('n= ');read(n); sn:=0 ; for i:=1 to n-1 do sn:=sn+u1*exp((i)*ln(q)); writeln('sn=' ,sn:8:12);readln; readln; End.

២.៤.េាាតកកណដាមបណតេាគកណែណកណបពបបកណធែណមាណ

គឲរ 1 ( 1)na a n d= + − ជ តនននយ ន 11

nnb b q −= × ជ តធែណមត រn∈ )ល

គណន ( ) 1 1 2 21

....n

n i i n ni

S a b a b a b a b=

= = + + +∑ ល

លមវធគណនផល បលនន var n,i:integer; a1,b1,q,d,sn:real; begin write('a1= ');read(a1); write('d= ');read(d); write('b1= ');read(b1); write('q= ');read(q); write('n= ');read(n); sn:=0 ; for i:=0 to n-1 do sn:=sn+(a1+i*d)*(b1*exp((i)*ln(q))); writeln('sn=' ,sn:8:12);readln; readln; End.

២.៥.េាាតកកណចបតបពណេផស ចបែែែលមវធគណនផល បល តន សែ េមរ

ល.( )

1 2 3 ....2! 3! 4! 1 !n

nSn

= + + + ++

. 1! 2! 3! .... !nS n= + + + +

គ. 2 2 2 21 1 1 1....1 2 3nS

n= + + + +



ឃ. 1 3 5 .... 2 1nS n= + + + + −

. 1 2 31 3 5 2 1...

2 1 2 1 2 1 2 1n nnS −

= + + + ++ + + +

ច. 2 31 3 5 2 1...2 2 2 2n n

nS −= + + + +

ឆ.2 3

1 ....2 6 !

n

nx x xS x

n= + + + + +

ជ.2 3 4 2

...2 3 4 2

n

nx x x xS x

n= − + − + +

ឈ.( )

3 5 7 2 1

....6 120 5040 2 1 !

n

nx x x xS x

n

−

= − + − + +−

ញ. 4 2 4 2 4 2 4 21 2 3 ...

1 1 1 2 2 1 3 3 1 1nnS

n n= + + + +

+ + + + + + + +

ន. 2 2 2 2sin 1 sin 2 sin 3 ... sin ( )nS n= + + + +

ថ. 2 3 4 5 11 2 3 4 ... nnS n += + + + + +

ឌ. 2 2 2 21! 2! 3! !...

ln 2 ln 3 ln 4 ln ( 1)nnSn

= + + + ++

ឍ.2 3

2 4 8 2

1 ...1 1 1 1 1n

n

nx x x xS

x x x x x= + + + + +

+ + + + +

ណ. 1! 1 2! 2 3! 3 ... !nS n n= + + + + ល

២.៦.េាគកណកណចបតបពណេផស

ចបែែែលមវធគណនផលគណៃន តេមរ

ល.3 3 3 3

3 3 3 32 1 3 1 4 1 1...2 1 3 1 4 1 1n

nPn

− − − −= × × × ×

+ + + +

. 2 2 2 21 1 1 11 1 1 .... 12 3 4nP

n = − − − −

គ. 2 31 3 5 2 1....

2 1 2 1 2 1 2 1n nnP −

= × × × ×+ + + +

ឃ. 1 3 5 2 1....2 4 6 2 2n

nPn+

= × × × ×+

. cos cos cos ....cos2 4 2n nx x xP x=



៣.ណតវn ៃបកណទមែ តាមបកណណ ប

៣.១.តទn ៃនសតែដលមនរបមនកណណ ន 1n nu au b+ = +

លមវធគណន var n,i:integer; a,b,u1,un:real; begin write('u1= ');read(u1); write('a= ');read(a); write('b= ');read(b); write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=a*un+b; writeln('un=' ,un:8:12);readln; End.

៣.២.តទn ៃនសតែដលមនរបមនកណណ ន 1n

nn

au bucu d+

+=

+

លមវធគណន var n,i:integer; a,b,c,d,u1,un:real; begin write('u1= ');read(u1); write('a= ');read(a); write('b= ');read(b); write('c= ');read(c); write('d= ');read(d); write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=(a*un+b)/(c*un+d); writeln('un=' ,un:8:12);readln; readln; End.

៣.៣.តទn ៃនសតែដលមនរបមនកណណ ន2

1n n nu au bu c+ = + +

លមវធគណន var n,i:integer; a,b,c,u1,un:real; begin write('u1= ');read(u1); write('a= ');read(a); write('b= ');read(b); write('c= ');read(c); write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=a*sqr(un)+b*un+c;



writeln('un=' ,un:8:12);readln; readln; End.

៣.៤.តទn ៃនសតែដលមនរបមនកណណ ន3 2

1n n n nu au bu cu d+ = + + +

លមវធគណន var n,i:integer; a,b,c,d,u1,un:real; begin write('u1= ');read(u1); write('a= ');read(a); write('b= ');read(b); write('c= ');read(c); write('d= ');read(d); write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=a*un*sqr(un)+b*sqr(un)+c*un+d; writeln('un=' ,un:8:12);readln; readln; End.

៣.៥.តទn ៃនសតែដលមនរបមនកណណ ន2

1 1 11

2 2

n nn

n

a u b u cua u b+

+ +=

+

លមវធគណន var n,i:integer; a1,b1,c1,a2,b2,u1,un:real; begin write('u1= ');read(u1); write('a1= ');read(a1); write('b1= ');read(b1); write('c1= ');read(c1); write('a2= ');read(a2); write('b2= ');read(b2); write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=(a1*sqr(un)+b1*un+c)/(a2*un+b2); writeln('un=' ,un:8:12);readln; readln; End.

៣.៦.តទn ៃនសតែដលមនរបមនកណណ ន2

1 1 11 2

2 2 2

n nn

n n

a u b u cua u b u c+

+ +=

+ +

var n,i:integer; a1,b1,c1,a2,b2,c2,u1,un:real; begin write('u1= ');read(u1); write('a1= ');read(a1);



write('b1= ');read(b1); write('c1= ');read(c1); write('a2= ');read(a2); write('b2= ');read(b2); write('c2= ');read(c2); write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=(a1*sqr(un)+b1*un+c)/(a2*un*un+b2*un+c2); writeln('un=' ,un:8:12);readln; readln; End.

៣.៧.តទn ៃនសតែដលមនរបមនកណណ ន 2 1n n nu au bu+ += +

លមវធគណន var n:integer; a,b,u1,u2:real; function un(n:integer):real; begin if n<1 then writeln('impossible!!! you should input n>0') else if n=1 then un:=u1 else if n=2 then un:=u2 else un:=a*un(n-1)+b*un(n-2); end; begin write('input a=');read(a); write('input b=');read(b); write('input u1=');read(u1); write('input u2=');read(u2); write('input n=');read(n); writeln('un= ' ,un(n):8:12);readln; readln; end.

៣.៨.តទn ៃនសតែដលមនរបមនកណណ ន 2 1n n nu au bu c+ += + +

លមវធគណន var n:integer; a,b,c,u1,u2:real; function un(n:integer):real; begin if n<1 then writeln('impossible!!! you should input n>0') else if n=1 then un:=u1 else if n=2 then un:=u2 else un:=a*un(n-1)+b*un(n-2)+c; end; begin write('input a=');read(a); write('input b=');read(b); write('input c=');read(c); write('input u1=');read(u1);



write('input u2=');read(u2); write('input n=');read(n); writeln('un= ' ,un(n):8:12);readln; readln; end.

៣.៩.តទn ៃនសតែដលមនរបមនកណណ ន2

1n n n nu u u au b+ = + + +

លមវធគណន var n,i:integer; a,b,u1,un:real; begin write('u1= ');read(u1); write('a= ');read(a); write('b= ');read(b); write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=un+sqrt(sqr(un)+b*un+c) writeln('un=' ,un:8:12);readln; readln; End.

៣.១០.តទn ៃនសតែដលមនរបមនកណណ ន 1n nu au b+ = +

លមវធគណន var n,i:integer; a,b,u1,un:real; begin write('u1= ');read(u1); write('a= ');read(a); write('b= ');read(b); write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=sqrt(a*un+b) writeln('un=' ,un:8:12);readln; readln; End.

៣.១១.តទn ៃនសតែដលមនរបមនកណណ ន2

1 2n nu u+ = −

លមវធគណន var n,i:integer; u1,un:real; begin write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=sqr(un)-2; writeln('un=' ,un:8:12);readln; readln; End.



៣.១២.តទn ៃនសតែដលមនរបមនកណណ ន3

1 3n n nu u u+ = −

លមវធគណន var n,i:integer; u1,un:real; begin write('n= ');read(n); un:=u1 ; for i:=1 to n-1 do un:=un*un*un-3*un; writeln('un=' ,un:8:12);readln; readln; End.





៤.អបកគមបាណេណមាណ

៤.១.ដយស ល sina គណនតៃមម cos ,tan ,cota a a

var sina,cosa,tana,cota:real;

begin

write('input sina= ');read(sina);

cosa:=sqrt(1-sqr(sina));

tana:=(sina)/(cosa);

cota:=1/(tana);

writeln('cosa=' ,cosa:8:12);readln;

writeln('tana=' ,tana:8:12);

writeln('cota=' ,cota:8:12);readln;

End.

៤.២.ដយស លcosa គណនតៃមម sin ,tan ,cota a a


begin

write('input cosa= ');read(cosa);

sina:=sqrt(1-sqr(cosa));

tana:=(sina)/(cosa);

cota:=1/(tana);

writeln('sina=' ,sina:8:12);readln;

writeln('tana=' ,tana:8:12);


End.

៤.៣.ដយស ល tan a គណនតៃមម sin ,cos ,cota a a


begin

write('input tana= ');read(tana);

cosa:=1/sqrt(1+sqr(tana));

sina:=tana*cosa;

cota:=1/(tana);




writeln('tana=' ,sina:8:12);


End.

៤.៤.ដយស ល cot a គណនតៃមម sin ,cos ,tana a a


begin

write('input cota= ');read(cota);

sina:=1/sqrt(1+sqr(cota));

sina:=tana*cosa;

cota:=1/(tana);


writeln('tana=' ,sina:8:12);


End.

៤.៥.ដយស ល tan2at = គណនតៃមម sin ,cos ,tan ,cota a a a

var sina,cosa,tana,cota,t:real;

begin

write('input tan(a/2)=t= '); read(t);

sina:=(2*t)/(1+t*t);

cosa:=(1-t*t)/(1+t*t);

tana:=(2*t)/(1-t*t);

cota:=(1-t*t)/(2*t);

writeln('sina=' , sina:8:12);readln;

writeln('cosa=' ,cosa:8:12);

writeln('tana= ',tana:8:12);


End.

៤.៦.ដយស ល sina គណន sin3a

var sina,sin3a:real;

begin

write('input sina= '); read(sina);

sin3a:=3*sina-4*sina*sina*sina;

writeln('sin3a=' , sin3a:8:12);readln;



readln;

End.

៤.៧.ដយស ល cosa គណន cos3a

var cosa,cos3a:real;

begin

write('input cosa= '); read(cosa);

cos3a:=4*cosa*cosa*cosa-3*cosa;

writeln('cos3a=' , cos3a:8:12);readln;

readln;

End.

៤.៨.ដយស ល tan a គណន tan3a

var tana,tan3a:real;

begin

write('input tana= '); read(tana);

tan3a:=(3*tana-tana*tana*tana)/(1-3*tana*tana);

writeln('tan3a=' , tan3a:8:12);readln;

readln;

End.

៤.៩.ដយស ល cot a គណន cot3a

var cota,tan3a,cot3a:real;

begin

write('input cota= '); read(cota);

tan3a:=(3*tana-tana*tana*tana)/(1-3*tana*tana);

cot3a:=1/tan3a;

writeln('cot3a=' , cot3a:8:12);readln;

readln;

End.



៥.ាណេណបែ ស

៥.១.ដយស លងជងបៃនតណ ABC គណនៃផផកររបសប

គឲរតេណ ABC មដមនជរ , ,BC a AC b AB c= = = ល

ចបែគណនៃផលកៃនតេណ ABC ល

នមគណនដ ែបមនយែ ហ ( )( )( )S p p a p b p c= − − − ងនល2

a b cp + +=

កមវធគណន

var a,b,c,p,S:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; S:=sqrt(p*(p-a)*(p-b)*(p-c)); writeln('S= ' ,S:8:12);readln; End.

៥.២.ដយស លងជងបៃនតណ ABC គណនររ សរងង រកៃនតណ

គឲរតេណ ABC មដមនជរ , ,BC a AC b AB c= = = ល ចបែគណនែង េែ ាែ ចលកតេណននល

ែបមនយ 4 ( )( )( )abc abcR

S p p a p b p c= =

− − − ល

លមវធគណន var a,b,c,p,S,R:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; S:=sqrt(p*(p-a)*(p-b)*(p-c)); R:=(a*b*c)/(4*S); writeln('R= ' ,R:8:12);readln; End.

៥.៣.ដយស លងជងបៃនតណ ABC គណនររ សរងង រកករងៃនតណ

គឲរតេណ ABC មដមនជរ , ,BC a AC b AB c= = = ល ចបែគណនែង េៃនែ ាែ ចលលគៃនតេណល

A

B Ca

bcS

A

B C

O

M

R



ែបមនយងនល ( )( )( )S p a p b p crp p

− − −= =

លមវធគណន var a,b,c,p,S,r:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; S:=sqrt(p*(p-a)*(p-b)*(p-c)); r:=S/p; writeln('r= ' ,r:8:12);readln; End.

៥.៤.ដយស លងជងបៃនតណ ABC គណន sin , sin , sinA B C


ចបែគណនsin , sin , sinA B C ?

លចយលនប 2sin sin sin

a b c RA B C= = =

លមវធគណន var a,b,c,p,S,R,sinA,sinB,sinC:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; S:=sqrt(p*(p-a)*(p-b)*(p-c)); R:=(a*b*c)/(4*S); sinA:=a/(2*R); sinB:=b/(2*R); sinC:=c/(2*R); writeln('sinA= ' ,sinA:8:12);readln; writeln('sinB= ' ,sinB:8:12); writeln('sinC= ' ,sinC:8:12);readln; End.

៥.៥.ដយស លងជងបៃនតណ ABC គណន cos , cos , cosA B C


ចបែគណនcos , cos , cosA B C ល

លចយលលបនប

2 2 2

2 2 2

2 2 2

2 cos2 cos2 cos

a b c bc Ab a c ac Bc a b ab A

= + −

= + − = + −

r

I

A

CB K



កមវធគណនា

var a,b,c,cosA,cosB,cosC:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); cosA:=(sqr(b)+sqr(c)-sqr(a))/(2*b*c); cosB:=(sqr(a)+sqr(c)-sqr(b))/(2*a*c); cosC:=(sqr(b)+sqr(a)-sqr(c))/(2*b*a); writeln('cosA= ' ,cosA:8:12);readln; writeln('cosB= ' ,cosB:8:12); writeln('cosC= ' ,cosC:8:12);readln; End.

៥.៦.ដយស លងជងបៃនតណ ABC គណន tan , tan , tanA B C

គឲរតេណ ABC មដមនជរ , ,BC a AC b AB c= = = ល ចបែគណន tan , tan , tanA B C ល

ដ ែបមនយ 2 2 2sin 4tancos

A SAA b c a

= =+ −

រន 2 2 21 sin , cos

2 2b c aS bc A A

bc+ −

= = ល


var a,b,c,p,S,tanA,tanB,tanC:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; S:=sqrt(p*(p-a)*(p-b)*(p-c)); tanA:=(4*S)/(sqr(b)+sqr(c)-sqr(a)); tanB:=(4*S)/(sqr(a)+sqr(c)-sqr(b)); tanC:=(4*S)/(sqr(a)+sqr(b)-sqr(c)); writeln('tanA= ' ,tanA:8:12);readln; writeln('tanB= ' ,tanB:8:12); writeln('tanC= ' ,tanC:8:12);readln; End.

៥.៧.ដយស លងជងបៃនតណ ABC គណន cot , cot , cotA B C

គឲរតេណ ABC មដមនជរ , ,BC a AC b AB c= = = ល ចបែគណនcot , cot , cotA B C ល

ែបមនយ 2 2 2 2 2 2 2 2 2

cot , cot ,cot4 4 4

b c a a c b a b cA B CS S S

+ − + − + −= = = ល




var a,b,c,p,S,cotA,cotB,cotC:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; S:=sqrt(p*(p-a)*(p-b)*(p-c); cotA:=(sqr(b)+sqr(c)-sqr(a))/(4*S)); cotB:=(sqr(a)+sqr(c)-sqr(b))/(4*S); cotC:=(sqr(a)+sqr(b)-sqr(c))/(4*S); writeln('cotA= ' ,cotA:8:12);readln; writeln('cotB= ' ,cotB:8:12); writeln('cotC= ' ,cotC:8:12);readln; End.

៥.៨.ដយស លងជងបៃនតណ ABC គណនររ សរងង រកករងមៃនតណ


ចបែគណនែង េែ ាែ ចលលគម , ,A B C ល

ែបមនយ ( )( )tan , ,2a b cA p p b p c S S Sr p r r

p a p a p b p c− −

= = = = =− − − −


var a,b,c,p,S,ra,rb,rc:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; S:=sqrt(p*(p-a)*(p-b)*(p-c)); ra:=S/(p-a); rb:=S/(p-b); rc:=S/(p-c); writeln('ra= ' ,ra:8:12);readln; writeln('rb= ' ,rb:8:12); writeln('rc= ' ,rc:8:12);readln; End.

A

B C

I

aI

bI

cI

1A

2B

2A2B1C

2C

ar

brcr



៥.៩.ដយស លងជងបៃនតណ ABC គណនររ សមដមនជងបៃនតណ


ចបែគណនែង មនដនយ ៃនតេណ ABC ល

លចយលមនដន

2 2 22

2 2 22

2 2 22

2 4

2 4

2 4

a

b

c

b c am

a c bm

a b cm

+= −

+ = −

+

= −

លមវធគណន var a,b,c,ma,mb,mc:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); ma:=sqrt((b*b+c*c)/2-(a*a)/4); mb:=sqrt((a*a+c*c)/2-(b*b)/4); mc:=sqrt((a*a+b*b)/2-(c*c)/4); writeln('ma= ' ,ma:8:12);readln; writeln('mb= ' ,mb:8:12); writeln('mc= ' ,mc:8:12);readln; End.

៥.១០.ដយស លងជងបៃនតណ ABC គណនររ សកមកសជងបៃនតណ

គឲរតេណ ABC មដមនជរ , ,BC a AC b AB c= = = ល ចបែគណនែង លមម យ ែតេណល

ែបមនយៃផលក 1 1 12 2 2a b cS a h b h c h= × = × = × ល

លមវធគណន var a,b,c,p,S,ha,hb,hc:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; S:=sqrt(p*(p-a)*(p-b)*(p-c)); ha:=(2*S)/a; hb:=(2*S)/b; hc:=(2*S)/c; writeln('ha= ' ,ha:8:12);readln; writeln('hb= ' ,hb:8:12); writeln('hc= ' ,hc:8:12);readln; End.

A

'AB C

'C 'BG

am

bm cm

A

B C'A

'B

'C

H



៥.១១.ដយស លងជងបៃនតណ ABC គណនែវងវងអតពពមករងជងបៃនតណ


ចបែគណនែង អ តននមលគយ ែតេណល

ណរបមនា

2 2 2

1 , 1 , 1A B Ca b cbc ac ab

b c a c a b = − = − = − + + +


var a,b,c,la,lb,lc:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); la:=sqrt(b*c*(1-sqr(a/(b+c)))); lb:=sqrt(a*c*(1-sqr(b/(a+c)))); lc:=sqrt(a*b*(1-sqr(c/(a+b)))); writeln('la= ' ,la:8:12);readln; writeln('lb= ' ,lb:8:12); writeln('lc= ' ,lc:8:12);readln; End.

៥.១២.ដយស លងជងបៃនតណ ABC គណន , , , ,OG OI OH IH IG នងGH

ែដលO ជផផ តរងង រកករង I ផផ តរងង រកG ជទ ទមទននង H ជវរតសង ៃនតណ

គឲរតេណ ABC មដមនជរ , ,BC a AC b AB c= = = ល ត , , ,O I G H ែម គៗ ជផាតែ ាែ ចលលគ ផាតែ ាែ ចលក លជលមរននអែតបៃន ABC∆ ល ចបែគណនរ

, , , ,OI OG OH GH IG ន IH ល

ែបមនយងនលតតវ 2 2OI R Rr= − រលចយល Euler )

2 2 2

2

9a b cOG R + +

= − រលចយលLeibnitz )

( )2 2 2 23 9OH OG R a b c= = − + +

2 2 2

22 29

a b cGH OG R + += = −

( ) ( )

( )

3 3 3 32 5 133 3

9

a b c a b c abcIG

a b c

+ + − + + −=

+ +

2 22 4 cos cos cosIH r R A B C= −

A

'A

'B'C

B C

I•



ងនល 2 2 2

2cos cos cos 18

a b cA B CR

+ += − ល

កមវធគណនា

var a,b,c,p,ro,ri,s,OI,OG,OH,GH,IH,IG:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; s:=sqrt(p*(p-a)*(p-b)*(p-c)); ro:=(a*b*c)/(4*s); ri:=s/p; OI:=sqrt(ro*ro-2*ro*ri); OG:=sqrt(ro*ro-(a*a+b*b+c*c)/9); OH:=3*OG; GH:=2*OG; IG:=sqrt(((16*p*p*p)-5*(a*a*a+b*b*b+c*c*c)-39*a*b*c)/54*p); IH:=sqrt(2*ri*ri+4*ro*ro-(a*a+b*b+c*c)/2); writeln('OI= ' ,OI:8:12);readln; writeln('OG= ' ,OG:8:12); writeln('OH= ' ,OH:8:12); writeln('GH= ' ,GH:8:12); writeln('IG= ' ,IG:8:12); writeln('IH= ' ,IH:8:12);readln; End.

៥.១៣.ដយស លងជងបៃនតណ ABC គណន sin sin sinA B B+ +


ចបែគណនsin sin sinA B B+ + ?

ណរបមន sin sin sin2

a b c pA B CR R

+ ++ + = =


var a,b,c,p,R,s,sumsine:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; s:=sqrt(p*(p-a)*(p-b)*(p-c)); R:=(a*b*c)/(4*s); sumsine:=p/R; writeln('sinA+sinB+sinC= ' ,sumsine:8:12);readln; readln; End.



៥.១៤.ដយស លងជងបៃនតណ ABC គណនcos cos cosA B C+ +


ចបែគណនcos cos cosA B C+ + ។

ណរបមន cos cos cos 1 rA B CR

+ + = +


var a,b,c,p,ro,ri,s,sumcosine:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; s:=sqrt(p*(p-a)*(p-b)*(p-c)); ro:=(a*b*c)/(4*s); ri:=s/p; sumcosine:=1+(ri/ro); writeln('cosA+cosB+cosC= ' ,sumcosine:8:12);readln; readln; End.

៥.១៥.ដយស លងជងបៃនតណ ABC គណន tan tan tanA B C+ +


ចបែគណន tan tan tanA B C+ + ?

ណរបន 2 2 2 2 2 2 2 2 24 4 4tan tan tan S S SA B C

b c a c a b a b c+ + = + +

+ − + − + −


var a,b,c,p,s,sumtan:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; s:=sqrt(p*(p-a)*(p-b)*(p-c)); sumtan:=(4*s)*(1/(b*b+c*c-a*a)+1/(a*a+c*c-b*b)+1/(a*a+b+b-c*c)); writeln('tanA+tanB+tanC= ' ,sumtan:8:12);readln; readln; End.

៥.១៦.ដយស លងជងបៃនតណ ABC គណនcot cot cotA B C+ +


ចបែគណនcot cot cotA B C+ + ?



ណរបមន 2 2 2

cot cot cot4

a b cA B CS

+ ++ + =


var a,b,c,p,s,sumcot:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; s:=sqrt(p*(p-a)*(p-b)*(p-c)); sumcot:=(a*a+b*b+c*c)/(4*s); writeln('cotA+cotB+cotC= ' ,sumcot:8:12);readln; readln; End.

៥.១៧.ដយស លងជងបៃនតណ ABC គណន sin ,sin ,sin2 2 2A B C


ចបែគណនsin ,sin ,sin2 2 2A B C

?

ណរបមន

( )( )

( )( )

( )( )

sin2

sin2

sin2

p b p cAbc

p a p cBac

p a p bCab

− − = − − = − − =


var a,b,c,p,sa,sb,sc:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; sa:=sqrt(((p-b)*(p-c))/(b*c)); sb:=sqrt(((p-a)*(p-c))/(a*c)); sc:=sqrt(((p-a)*(p-b))/(a*b)); writeln('sinA/2= ' ,sa:8:12);readln; writeln('sinB/2= ' ,sb:8:12); writeln('sinC/2= ' ,sc:8:12);readln; readln; End.



៥.១៨.ដយស លងជងបៃនតណ ABC គណន cos ,cos ,cos2 2 2A B C


ចបែគណនcos ,cos ,cos2 2 2A B C

?

ណរបមន

( )

( )

( )

cos2

cos2

cos2

p p aAbc

p p bBac

p p cCab

− = − = − =

var a,b,c,p,ca,cb,cc:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; ca:=sqrt((p*(p-a))/(b*c)); cb:=sqrt((p*(p-b))/(a*c)); cc:=sqrt((p*(p-c))/(a*b)); writeln('cosA/2= ' ,ca:8:12);readln; writeln('cosB/2= ' ,cb:8:12); writeln('cosC/2= ' ,cc:8:12);readln; readln; End.

៥.១៩.ដយស លងជងបៃនតណ ABC គណន tan ,tan ,tan2 2 2A B C


ចបែគណន tan ,tan ,tan2 2 2A B C

ណរបមន

( )( )( )

( )( )( )

( )( )( )

tan2

tan2

tan2

p b p cAp p a

p a p cBp p b

p a p bCp p c

− − =

−

− −= −

− − = −




var a,b,c,p,ta,tb,tc:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; ta:=sqrt(((p-b)*(p-a))/(p*(p-a))); tb:=sqrt(((p-a)*(p-c))/(p*(p-b))); tc:=sqrt(((p-a)*(p-b))/(p*(p-c))); writeln('tanA/2= ' ,ta:8:12);readln; writeln('tanB/2= ' ,tb:8:12); writeln('tanC/2= ' ,tc:8:12);readln; readln; End.

៥.២០.ដយស លងជងបៃនតណ ABC គណន cot ,cot ,cot2 2 2A B C


ចបែគណនcot ,cot ,cot2 2 2A B C

?

ណរបមន

( )( )( )

( )( )( )

( )( )( )

cot2

cot2

cot2

p p aAp b p c

p p bBp a p c

p p cCp a p b

− =

− −

−= − −

− = − −


var a,b,c,p,coa,cob,coc:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; coa:=sqrt((p*(p-a))/((p-b)*(p-a))); cob:=sqrt((p*(p-b))/((p-a)*(p-c))); coc:=sqrt((p*(p-c))/((p-a)*(p-a))); writeln('cotA/2= ' ,coa:8:12);readln; writeln('cotB/2= ' ,cob:8:12); writeln('cotC/2= ' ,coc:8:12);readln; readln; End.



៥.២១.ដយស លងជងបៃនតណ ABC គណន tan tan tan2 2 2A B C+ +


ចបែគណន tan tan tan2 2 2A B C+ + ?

ណរបមន 4tan tan tan

2 2 2A B C r R

p+

+ + =


var a,b,c,p,s,ro,ri,sumtan:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; s:=sqrt(p*(p-a)*(p-b)*(p-c)); ro:=(a*b*c)/(4*s); ri:=s/p; sumtan:=(ri+4*ro)/p; writeln('tanA/2+tanB/2+tanC/2= ' ,sumtan:8:12);readln; readln; End.

៥.២២.ដយស លងជងបៃនតណ ABC គណន cot cot cot2 2 2A B C

+ +


ចបែគណនcot cot cot2 2 2A B C

+ + ?

ណរបមន cot cot cot2 2 2A B C p

r+ + =

var a,b,c,p,s,ri,sumcot:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; s:=sqrt(p*(p-a)*(p-b)*(p-c)); ri:=s/p; sumcot:=p/ri; writeln('cotA/2+cotB/2+cotC/2= ' ,sumcot:8:12);readln; End.

៥.២៣.ដយស លងជងបៃនតណ ABC គណនររ សម Brocard


ចបែគណនែង មω ៃនចណច Brocard លគតេណ ABC ល



P ជចណច Brocard នលគតេណ ABC ងនល PAB PBC PCA ω∠ = ∠ = ∠ = ល

ែបមនយ 2 2 2

cot cot cot cot4

a b cA B CS

ω + += + + =

នន 2 2 24tan SArc

a b cω = + +

ល

កមវធ

var a,b,c,p,s,R,brocardangle:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); p:=(a+b+c)/2; s:=sqrt(p*(p-a)*(p-b)*(p-c)); R:=(a*b*c)/(4*S); brocardangle:=Arctan((4*s)/(a*a+b*b+c*c)); writeln('Brocard Angle= ' ,brocardangle:8:12);readln; readln; End.


P

A

B C

•

•




៦.ចណកេណត

៦.១.ៃេាកកចណកេណត ដោេ ាសារជបេាាតកមកវាសកពតាពែរ គឲរចតេណេហ ABCD មដមនជរ , ,AB a BC b CD c= = = ន DA d=

ដផល បលមឈម A នC ឬ B ន D នច2u ល គណនៃផលកចតេណននណ

ែបមនយ ( )( )( )( ) 2cosS p a p b p c p d abcd u= − − − − −

ងនល 2

a b c dp + + += ន cos cos cos

2 2A C B Du + +

= = ល


var a,b,c,d,p,s,u:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); write('input d= ');read(d); write('input u= ');read(u); p:=(a+b+c+d)/2; s:=sqrt((p-a)*(p-b)*(p-c)*(p-d)-a*b*c*d*cos(u)*cos(u)); writeln('S= ' ,s:8:12);readln; End.

៦.២.ៃផផករឡតណណ ងដយស លែវងងនងររ សមទលកពលពរ

គឲរចតេណេហ ABCD មដមនជរ , ,AB a BC b CD c= = = ន DA d= ដគនចម A នC រឬ B ន D ) គណនៃផលកចតេណននណ

ែបមនយ ( )( )( )( ) 2cos2

A CS p a p b p c p d abcd += − − − − −

ងនល 2

a b c dp + + += ល

A

B

C

D

a

b

c

d




var a,b,c,d,p,s,u,A1,C1:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); write('input d= ');read(d); write('input A= ');read(A1); write('input C= ');read(C1); u:=(A1+C1)/2; p:=(a+b+c+d)/2; s:=sqrt((p-a)*(p-b)*(p-c)*(p-d)-a*b*c*d*cos(u)*cos(u)); writeln('S= ' ,s:8:12);readln; readln; End. ៦.៣.ៃផផករឡតណណ ងដយស លែវងវងអតទអងជងពរនងមរបងវងអតជងពរ

គឲរចតេណេហ ABCD មដមនអ តលត 1AC d= ន 2BD d= ដមផ េដ អ តលតនែននគគu ល គណនៃផលកចតេណនន ល

ែបមនយ 1 21 sin2

S d d u= × × ល


var d1,d2,s,u,v:real; begin write('input d1= ');read(d1); write('input d2= ');read(d2); write('input u= ');read(u); v:=sin(u); s:=(d1*d2*v)/2; writeln('s= ' ,s:8:12);readln; readln; End.

u

A

B

C

D

1d

•2d



៧.ចណកេណត ែ ចកកៗក ែស

៧.១.ៃផផករឡតណណ ងរកករងរងង ដយស លែវងងជងបន

គឲរចតេណេហ ABCD មដមនជរ , ,AB a BC b CD c= = = ន DA d=

ែបមនយ ( )( )( )( )S p a p b p c p d= − − − − ងនល 2

a b c dp + + += ល

កមវធ

var a,b,c,d,p,s:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); write('input d= ');read(d); p:=(a+b+c+d)/2; s:=sqrt((p-a)*(p-b)*(p-c)*(p-d)); writeln('s= ' ,s:8:12);readln; readln; End. ៧.២.ផលគណវងអតទអងជងពរៃនឡតណណ ងរកករងរងង ដយស លែវងងជងបន

A

B

C

D

a

d

b

c

A

B

C

D

a

d

b

c1d

2d



គឲរចតេណេហ ABCD មដមនជរ , ,AB a BC b CD c= = = ន DA d= ល ចបែគណនផលគណអ តលតយ នែ ល

របមន 1 2d d ac bd× = + (ទសបទតលមម


var a,b,c,d,d1d2:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); write('input d= ');read(d); d1d2:=(a*c+b*d); writeln('d1.d2= ' ,d1d2:8:12);readln; readln; End.

៧.៣.ផលធៀបវងអតទអងជងពរៃនឡតណណ ងរកករងរងង ដយស លែវងងជងបន

គឲរចតេណេហ ABCD មដមនជរ , ,AB a BC b CD c= = = ន DA d= ល ចបែគណនផលធមអ តលតៃនចតេណ ABCD ល var a,b,c,d,d0:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); write('input d= ');read(d); d0:=(a*d+b*c)/(a*b+c*d); writeln('d1/d2= ' ,d0:8:12);readln; readln; End.

៧.៤.ែវងវងអតទអងជងពរៃនឡតណណ ងរកករងរងង ដយស លែវងងជងបន

គឲរចតេណេហ ABCD មដមនជរ , ,AB a BC b CD c= = = ន DA d= ល ចបែគណនងវអ តលតយ នែៃនចតេណ ABCD ល


var a,b,c,d,d1,d2:real; begin a:=3900;b:=5200;c:=2500;d:=6000; write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); write('input d= ');read(d); d1:=sqrt(((a*c+b*d)*(a*d+b*c))/(a*b+c*d)); d2:=sqrt(((a*b+c*d)*(a*c+b*d))/(a*d+b*c)); writeln('d1= ' ,d1:8:12);readln; writeln('d2= ' ,d2:8:12);readln; End.



៨.កតអែដបកៗក ាា ស

៨.១.កវរដនឡណឡកក លៃនវងអតម

គឲរនែចណច ( ),a aA x y ន ( ),b bB x y ល

ចបែែែលមវធគណនលបអែេនចណចលចយ លៃនអ ត AB ល

ែបមនយ ,2 2

a b a bi i

x x y yI x y+ + = =

ល

កមវធគណនរ var xa,ya,xb,yb,xi,yi:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input xb= ');read(xb); write('input yb= ');read(yb); xi:=(xa+xb)/2; yi:=(ya+yb)/2; writeln('xi= ' ,xi:8:12);readln; writeln('yi= ' ,yi:8:12);readln; End.

៨.២.ឡមទ របងពរឡណឡករងមង

គឲរនែចណច ( ),a aA x y ន ( ),b bB x y ល

ចបែែែលមវធគណនចមរ ដែយនែចណច A ន B ល

ែបមនយ ( ) ( )2 2b a b aAB x x y y= − + − ល

លមវធគណនរ var xa,ya,xb,yb,AB:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input xb= ');read(xb); write('input yb= ');read(yb); AB:=sqrt(sqr(xb-xa)+sqr(yb-ya)); writeln('AB= ' ,AB:8:12);readln; readln; End.



៨.៣.កវរដនៃនឡណឡែឡកករងវងអតម

គឲរនចណច ( ),a aA x y ន ( ),b bB x y ល ចបែែែលមវធគណនលបអែេនៃនចណច

P ងចលលគអ ត AB តមផលធម mn

ល

ែបមនយ ;b a b ap p

mx nx my nyx ym n m n+ +

= =+ +

ល


var xa,ya,xb,yb,m,n,xp,yp:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input xb= ');read(xb); write('input yb= ');read(yb); write('input m= ');read(m); write('input n= ');read(n); xp:=(m*xb+n*xa)/(m+n); yp:=(m*yb+n*ya)/(m+n); writeln('xp= ' ,xp:8:12);readln; writeln('yp= ' ,yp:8:12); readln; End.

៨.៤.កវរដនៃនឡណឡែឡកវងអតម

គឲរនចណច ( ),a aA x y ន ( ),b bB x y ល ចបែែែលមវធគណនលបអែេនៃនចណច

Q ងចលកអ ត AB តមផលធម mn

ល

ែបមនយ ;b a b aq q

mx nx my nyx ym n m n− −

= =− −

ល


var xa,ya,xb,yb,m,n,xq,yq:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input xb= ');read(xb); write('input yb= ');read(yb); write('input m= ');read(m); write('input n= ');read(n); xq:=(m*xb-n*xa)/(m-n); yq:=(m*yb-n*ya)/(m-n); writeln('xq= ' ,xq:8:12);readln; writeln('yq= ' ,yq:8:12); readln; End.



៨.៥.កវរដនទ ទមទនៃនតណម

គឲរចណច ( ) ( ), , ,a a b bA x y B x y ន ( ),b bC x y តេនជតេណមដល

ចបែែែលមវធគណនលបអែេនលជលមរនៃនតេណ ABC ល

តមលនលលន 3

OA OB OCOG→ → →

→ + += នន 3

3

a b cg

a b cg

x x xx

y y yy

+ + =

+ + =

var xa,ya,xb,yb,xc,yc,xg,yg:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input xb= ');read(xb); write('input yb= ');read(yb); write('input xc= ');read(xc); write('input yc= ');read(yc); xg:=(xa+xb+xc)/3; yg:=(ya+yb+yc)/3; writeln('xg=' ,xg:8:12);readln; writeln('yg=' ,yg:8:12);readln; End.

៨.៦.ែវងងៃនតណមដយស លកវរដនកពល


ចបែែែលមវធគណនែង ជរយ ៃនតេណ ABC ល

ែបមនយចមរ ដែយនែចណចលគ ថ ( ) ( )2 2b a b aAB x x y y= − + − ល


var xa,ya,xb,yb,xc,yc,a,b,c:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input xb= ');read(xb); write('input yb= ');read(yb); write('input xc= ');read(xc); write('input yc= ');read(yc); a:=sqrt(sqr(xc-xb)+sqr(yc-yb)); b:=sqrt(sqr(xc-xa)+sqr(yc-ya)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)); writeln('a=' ,a:8:12);readln; writeln('b=' ,b:8:12); writeln('c=' ,c:8:12);readln; End.

A

'AB C

'C 'BG



៨.៧.កវរដនផផ តរងង រកករងតណម


ចបែែែលមវធគណនលបអែេនផាតែ ាែ ចលលគៃនតេណ ABC ល

កមវធគណនកវរដនផផ តរងង រកករងតណ :ABC

var xa,ya,xb,yb,xc,yc,a,b,c,xi,yi:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input xb= ');read(xb); write('input yb= ');read(yb); write('input xc= ');read(xc); write('input yc= ');read(yc); a:=sqrt(sqr(xc-xb)+sqr(yc-yb)); b:=sqrt(sqr(xc-xa)+sqr(yc-ya)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)); xi:=(a*xa+b*xb+c*xc)/(a+b+c); yi:=(a*ya+b*yb+c*yc)/(a+b+c); writeln('xi=' ,xi:8:12);readln; writeln('yi=' ,yi:8:12); readln; End.

៨.៨.ៃផផករតណមយស លកវរដនកពល


ចបែែែលមវធគណនៃផលកៃនតេណ ABC ល

ែបមនយ 1 1det ,2 2

b a b a

c a c a

x x y yS AB AC

x x y y→ → − −

= = − −

( )( ) ( )( )12 b a c a c a b ax x y y x x y y= − − − − −

លមវធគណនៃផលក var xa,ya,xb,yb,xc,yc,S:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input xb= ');read(xb); write('input yb= ');read(yb); write('input xc= ');read(xc); write('input yc= ');read(yc); S:=abs((xb-xa)*(yc-ya)-(xc-xa)*(yc-ya))/2; writeln('S=' ,S:8:12);readln; readln; End.



៨.៩.ររ សរងង រកករងតណមដយស លកវរដនកពល

var xa,ya,xb,yb,xc,yc,a,b,c,R,S:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input xb= ');read(xb); write('input yb= ');read(yb); write('input xc= ');read(xc); write('input yc= ');read(yc); a:=sqrt(sqr(xc-xb)+sqr(yc-yb)); b:=sqrt(sqr(xc-xa)+sqr(yc-ya)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)); S:=abs((xb-xa)*(yc-ya)-(xc-xa)*(yc-ya))/2; R:=(a*b*c)/(4*S); writeln('R=' ,R:8:12);readln; readln; End.

៨.១០.ររ សរងង រកតណមដយស លកវរដនកពល

var xa,ya,xb,yb,xc,yc,a,b,c,r,p,S:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input xb= ');read(xb); write('input yb= ');read(yb); write('input xc= ');read(xc); write('input yc= ');read(yc); a:=sqrt(sqr(xc-xb)+sqr(yc-yb)); b:=sqrt(sqr(xc-xa)+sqr(yc-ya)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)); p:=(a+b+c)/2; S:=abs((xb-xa)*(yc-ya)-(xc-xa)*(yc-ya))/2; r:=S/p; writeln('r=' ,r:8:12);readln; readln; End.

៨.១១.កវរដនៃនផផ តរងង រកតណម

លមវធគណនលបអែេន ( )0 0,O x y ផាតែ ាែ ចលកតេណ រ

A

B C

O



var xa,ya,xb,yb,xc,yc,xo,yo,p,q,u,v:real; begin write('input xa=');read(xa); write('input ya=');read(ya); write('input xb=');read(xb); write('input yb=');read(yb); write('input xc=');read(xc); write('input yc=');read(yc); xa:=-2;ya:=9;xb:=0;yb:=3;xc:=2;yc:=1; p:=(xc*xc-xa*xa)*(yb-ya)-(xb*xb-xa*xa)*(yc-ya)+(yb-ya)*(yc-yb)*(yc-ya); q:=(xc-xa)*(yb-ya)-(xb-xa)*(yc-ya); u:=(xb-xa)*(yc*yc-ya*ya)-(xc-xa)*(yb*yb-ya*ya)+(xb-xa)*(xc-xb)*(xc-xa); v:=(xb-xa)*(yc-ya)-(xc-xa)*(yb-ya); xo:=p/(2*q); yo:=u/(2*v); writeln(' xo= ',xo:8:12); readln; writeln(' yo= ' ,yo:8:12); readln; End.

៨.១២.កវរដនវរតសង ៃនតណម

លមវធគណនលបអែេន ( , )h hH x y អែតបៃនតេណ រ

var xa,ya,xb,yb,xc,yc,xo,yo,xh,yh,p,q,u,v:real; begin write('input xa=');read(xa); write('input ya=');read(ya); write('input xb=');read(xb); write('input yb=');read(yb); write('input xc=');read(xc); write('input yc=');read(yc); p:=(xc*xc-xa*xa)*(yb-ya)-(xb*xb-xa*xa)*(yc-ya)+(yb-ya)*(yc-yb)*(yc-ya); q:=(xc-xa)*(yb-ya)-(xb-xa)*(yc-ya); u:=(xb-xa)*(yc*yc-ya*ya)-(xc-xa)*(yb*yb-ya*ya)+(xb-xa)*(xc-xb)*(xc-xa); v:=(xb-xa)*(yc-ya)-(xc-xa)*(yb-ya); xo:=p/(2*q); yo:=u/(2*v);

H

A

B C'A

'B'C



xh:=xa+xb+xc-2*xo; yh:=ya+yb+yc-2*yo; writeln(' xh= ',xh:8:12); readln; writeln(' yh= ' ,yh:8:12); readln; end.

៨.១៣.កវរដនៃន , , ,G O H I


var xa,ya,xb,yb,xc,yc,xo,yo,xh,yh,xi,yi,xg,yg,u1,v1,u2,v2,a,b,c:real; begin write('input xa=');read(xa); write('input ya=');read(ya); write('input xb=');read(xb); write('input yb=');read(yb); write('input xc=');read(xc); write('input yc=');read(yc); u1:=(xc*xc-xa*xa)*(yb-ya)-(xb*xb-xa*xa)*(yc-ya)+(yb-ya)*(yc-yb)*(yc-ya); v1:=(xc-xa)*(yb-ya)-(xb-xa)*(yc-ya); u2:=(xb-xa)*(yc*yc-ya*ya)-(xc-xa)*(yb*yb-ya*ya)+(xb-xa)*(xc-xb)*(xc-xa); v2:=(xb-xa)*(yc-ya)-(xc-xa)*(yb-ya); a:=sqrt(sqr(xb-xc)+sqr(yb-yc)); b:=sqrt(sqr(xc-xa)+sqr(yc-ya)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)); xo:=u1/(2*v1); yo:=u2/(2*v2); xh:=xa+xb+xc-2*xo; yh:=ya+yb+yc-2*yo; xi:=(a*xa+b*xb+c*xc)/(a+b+c); yi:=(a*ya+b*yb+c*yc)/(a+b+c); xg:=(xa+xb+xc)/3; yg:=(ya+yb+yc)/3; writeln(' xo= ',xo:8:12); readln; writeln(' yo= ',yo:8:12); writeln(' xh= ' ,xh:8:12); writeln(' yh= ',yh:8:12); writeln(' xi= ' ,xi:8:12); writeln(' yi= ' ,yi:8:12); writeln(' xg= ',xg:8:12); writeln(' yg= ' ,yg:8:12); readln; end.



៨.១៤.ែវង , , ,OG OI OH IH នង IG


var xa,ya,xb,yb,xc,yc,xo,yo,xh,yh,xi,yi,xg,yg,u1,v1,u2,v2,a,b,c, OI,OG,OH,IH,IG,GH:real; begin write('input xa=');read(xa); write('input ya=');read(ya); write('input xb=');read(xb); write('input yb=');read(yb); write('input xc=');read(xc); write('input yc=');read(yc); u1:=(xc*xc-xa*xa)*(yb-ya)-(xb*xb-xa*xa)*(yc-ya)+(yb-ya)*(yc-yb)*(yc-ya); v1:=(xc-xa)*(yb-ya)-(xb-xa)*(yc-ya); u2:=(xb-xa)*(yc*yc-ya*ya)-(xc-xa)*(yb*yb-ya*ya)+(xb-xa)*(xc-xb)*(xc-xa); v2:=(xb-xa)*(yc-ya)-(xc-xa)*(yb-ya); a:=sqrt(sqr(xb-xc)+sqr(yb-yc)); b:=sqrt(sqr(xc-xa)+sqr(yc-ya)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)); xo:=u1/(2*v1); yo:=u2/(2*v2); xh:=xa+xb+xc-2*xo; yh:=ya+yb+yc-2*yo; xi:=(a*xa+b*xb+c*xc)/(a+b+c); yi:=(a*ya+b*yb+c*yc)/(a+b+c); xg:=(xa+xb+xc)/3; yg:=(ya+yb+yc)/3; OI:=sqrt(sqr(xi-xo)+sqr(yi-yo)); OG:=sqrt(sqr(xg-xo)+sqr(yg-yo)); OH:=sqrt(sqr(xh-xo)+sqr(yh-yo)); IH:=sqrt(sqr(xh-xi)+sqr(yh-yi)); IG:=sqrt(sqr(xg-xi)+sqr(yg-yi)); GH:=sqrt(sqr(xh-xg)+sqr(yh-yg)); writeln(' xo= ',xo:8:12); readln; writeln(' yo= ',yo:8:12); writeln(' xh= ' ,xh:8:12); writeln(' yh= ',yh:8:12); writeln(' xi= ' ,xi:8:12); writeln(' yi= ' ,yi:8:12); writeln(' xg= ',xg:8:12); writeln(' yg= ' ,yg:8:12); writeln(' OI= ',OI:8:12); writeln(' OG= ',OG:8:12); writeln(' OH= ' ,OH:8:12); writeln(' IH= ',IH:8:12); writeln(' IG= ' ,IG:8:12); writeln(' GH= ' ,GH:8:12);readln; end.



៨.១៥.ែវងមដមនជងបៃនតណដយស លកវរដនកពល

គឲរតេណ ABC មដមនលនបល ( ) ( ), , ,a a b bA x y B x y ន ( ),c cC x y ល

ចបែគណនែង មនដនយ ៃនតេណ ABC ល

ត , ,K L M ជចណចលចយ លៃន , ,BC CA AB ែម គៗ ននមនដនយ គគ

( ) ( )

( ) ( )

( ) ( )

2 2

2 2

2 2

a k a k a

b l b l b

c m c m c

m AK x x y y

m BL x x y y

m CM x x y y

= = − + −

= = − + −

= = − + −

លមវធគណនមនដនយ រ var xa,ya,xb,yb,xc,yc,xk,yk,xl,yl,xm,ym,ma,mb,mc:real; begin write('input xa=');read(xa); write('input ya=');read(ya); write('input xb=');read(xb); write('input yb=');read(yb); write('input xc=');read(xc); write('input yc=');read(yc); xk:=(xb+xc)/2;yk:=(yb+yc)/2; xl:=(xa+xc)/2;yl:=(ya+yc)/2; xm:=(xa+xb)/2;ym:=(ya+yb)/2; ma:=sqrt(sqr(xk-xa)+sqr(yk-ya)); mb:=sqrt(sqr(xl-xb)+sqr(yl-yb)); mc:=sqrt(sqr(xm-xc)+sqr(ym-yc)); writeln(' ma= ',ma:8:12); readln; writeln(' mb= ',mb:8:12); writeln(' mc= ' ,mc:8:12);readln; end.

៨.១៦.ែវងកមកសជងបៃនត ណដយស លកវរដនកពល


ចបែគណនែង លមមយ ៃនតេណ ABC ល

ែបមនយ ( )( )( )1 1 1. . .2 2 2a b cS a h b h c h p p a p b p c= = = = − − − ល

លមវធគណនលមមយ var xa,ya,xb,yb,xc,yc,p,S,a,b,c,ha,hb,hc:real; begin write('input xa=');read(xa); write('input ya=');read(ya); write('input xb=');read(xb); write('input yb=');read(yb); write('input xc=');read(xc); write('input yc=');read(yc); a:=sqrt(sqr(xb-xc)+sqr(yb-yc));



b:=sqrt(sqr(xc-xa)+sqr(yc-ya)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)); p:=(a+b+c)/2; S:=sqrt(p*(p-a)*(p-b)*(p-c)); ha:=(2*S)/a; hb:=(2*S)/b; hc:=(2*S)/c; writeln('ha= ' ,ha:8:12);readln; writeln('hb= ' ,hb:8:12); writeln('hc= ' ,hc:8:12);readln; End.

៨.១៧.ែវងវងអតពពមករងជងបៃនតណដយស លកវរដនកពល


ចបែគណនែង អ តននមលគយ ៃនតេណ ABC ល

ជន បតតវគណនែង ជរយ ែតេណ

( ) ( )

( ) ( )

( ) ( )

2 2

2 2

2 2

c b c b

c a c a

b a b a

a BC x x y y

b AC x x y y

c AB x x y y

= = − + −

= = − + − = = − + −

ែបមនយ

2

2

2

1

1

1

a

b

c

al bcb c

bl aca c

cl aba b

= − + = − +

= − +

លមវធគណនែង អ តននលគៃនមយ ែតេណ var xa,ya,xb,yb,xc,yc,a,b,c,la,lb,lc:real; begin write('input xa=');read(xa); write('input ya=');read(ya); write('input xb=');read(xb); write('input yb=');read(yb); write('input xc=');read(xc); write('input yc=');read(yc); a:=sqrt(sqr(xb-xc)+sqr(yb-yc)); b:=sqrt(sqr(xc-xa)+sqr(yc-ya)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)); la:=sqrt((b*c)*(1-sqr(a/(b+c)))); lb:=sqrt((a*c)*(1-sqr(b/(a+c)))); lc:=sqrt((a*b)*(1-sqr(c/(a+b))));



writeln('la= ' ,la:8:12);readln; writeln('lb= ' ,lb:8:12); writeln('lc= ' ,lc:8:12);readln; End.

៨.១៨.ឡមទ ពកពលមេឡណឡែឡកករងៃនងជមរបសតណ


, ,K L M ជចណចងចលលគៃនអ ត , ,BC CA AB ែម គៗ តមផលធមរ

, ,KB LC MAu v wKC LA MB

= = = ល គណនែង , ,AK BL CM ល

ជន បដ តតវគណនលបអែេនៃនចណចងចលលគ , ,K L M តមែបមនយេមរ

, ; , , ,1 1 1 1 1 1b c b c c a c a a b a bx u x y uy x vx y vy x w x y wyK L M

u u v v w w+ + + + + +

+ + + + + +

ែបមនយគណនចមរ ដែយនែចណចល

កមវធសមបគណន , ,AK BL CM var xa,ya,xb,yb,xc,yc,a,b,c,xk,yk,xl,yl,xm,ym,u,v,w,AK,BL,CM:real; begin write('input xa=');read(xa); write('input ya=');read(ya); write('input xb=');read(xb); write('input yb=');read(yb); write('input xc=');read(xc); write('input yc=');read(yc); write('input u=');read(u); write('input v=');read(v); write('input w=');read(w); xk:=(xb+u*xc)/(1+u);yk:=(yb+u*yc)/(1+u); xl:=(xc+v*xa)/(1+v);yl:=(yc+v*ya)/(1+v); xm:=(xa+w*xb)/(1+w);ym:=(ya+w*yb)/(1+w); AK:=sqrt(sqr(xk-xa)+sqr(yk-ya)); BL:=sqrt(sqr(xl-xb)+sqr(yl-yb)); CM:=sqr(sqr(xm-xc)+sqr(ym-yc)); writeln('AK= ' ,AK:8:12);readln; writeln('BL= ' ,BL:8:12); writeln('CM= ' ,CM:8:12);readln; End.

៨.១៩.ឡមទ ពកពលមេឡណឡែឡកៃនងជមរបសតណ


, ,K L M ជចណចងចលកៃនអ ត , ,BC CA AB ែម គៗ តមផលធមរ

, ,KB LC MAu v wKC LA MB

= = = ល គណនែង , ,AK BL CM ល

ជន បដ តតវគណនលបអែេនៃនចណចងចលលគ , ,K L M តមែបមនយេមរ



, ; , , ,1 1 1 1 1 1b c b c c a c a a b a bx u x y uy x vx y vy x w x y wyK L M

u u v v w w− − − − − −

− − − − − −

ែបមនយគណនចមរ ដែយនែចណចល

កមវធសមបគណន , ,AK BL CM var xa,ya,xb,yb,xc,yc,a,b,c,xk,yk,xl,yl,xm,ym,u,v,w,AK,BL,CM:real; begin write('input xa=');read(xa); write('input ya=');read(ya); write('input xb=');read(xb); write('input yb=');read(yb); write('input xc=');read(xc); write('input yc=');read(yc); write('input u=');read(u); write('input v=');read(v); write('input w=');read(w); xk:=(xb-u*xc)/(1-u);yk:=(yb-u*yc)/(1-u); xl:=(xc-v*xa)/(1-v);yl:=(yc-v*ya)/(1-v); xm:=(xa-w*xb)/(1-w);ym:=(ya-w*yb)/(1-w); AK:=sqrt(sqr(xk-xa)+sqr(yk-ya)); BL:=sqrt(sqr(xl-xb)+sqr(yl-yb)); CM:=sqr(sqr(xm-xc)+sqr(ym-yc)); writeln('AK= ' ,AK:8:12);readln; writeln('BL= ' ,BL:8:12); writeln('CM= ' ,CM:8:12);readln; End.

៨.២០.ែវងរងង រកករងមៃនតណ


ចបែែែលមវធគណនែង េៃនែ លគមៃនតេណ ABC ល

ជន បតតវគណនែង ជរយ ៃនតេណ ABC

( ) ( )

( ) ( )

( ) ( )

2 2

2 2

2 2

c b c b

c a c a

b a b a

a BC x x y y

b AC x x y y

c AB x x y y

= = − + −

= = − + − = = − + −

គណន , ,a b cr r r តមែបមនយ រ

( )( )tan , ,2a b cA p p b p c S S Sr p r r

p a p a p b p c− −

= = = = =− − − −

លមវធគណន var xa,ya,xb,yb,xc,yc,a,b,c,p,S,ra,rb,rc:real; begin



write('input xa=');read(xa); write('input ya=');read(ya); write('input xb=');read(xb); write('input yb=');read(yb); write('input xc=');read(xc); write('input yc=');read(yc); a:=sqrt(sqr(xb-xc)+sqr(yb-yc)); b:=sqrt(sqr(xc-xa)+sqr(yc-ya)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)); p:=(a+b+c)/2; S:=sqrt(p*(p-a)*(p-b)*(p-c)); ra:=S/(p-a); rb:=S/(p-b); rc:=S/(p-c); writeln('ra= ' ,ra:8:12);readln; writeln('rb= ' ,rb:8:12); writeln('rc= ' ,rc:8:12);readln; End.

៨.២១.ឡមទ ពឡណឡមេបនផ តមករងមង

គឲរន ត ( ) : 0d ax by c+ + = នចណច ( )0 0,A x y ល

ចបែែែលមវធគណនចមរ ដនចណច A ទន ត ( )d ល

ែបមនយ ( ) 0 02 2

| |,( ) ax by cd A da b+ +

=+

ល

លមវធមគណន var a,b,c,xo,yo,d,d1,d2:real; begin write('input a= ');read(a); write('input b= ');read(b); write('input c= ');read(c); write('input xo= ');read(xo); write('input yo= ');read(yo); d:=abs(a*xo+b*yo+c)/sqrt(a*a+b*b); writeln('d(A,(d))= ',d:8:12);readln; readln; End.

៨.២២.ឡមទ របងបនផ តពរសប រ

គឲរន តនែ ( )1 1 1 1: 0d a x b y c+ + = ន ( )2 2 2 2: 0d a x b y c+ + = ល

1 1

2 2

a ba b

= ននន ត ( ) ( )1 2/ /d d លចបែែែលមវធគណនចមរ ដែយន តយ នែននល

var a1,b1,c1,a2,b2,c2,d:real; begin write('input a1= ');read(a1); write('input b1= ');read(b1); write('input c1= ');read(c1); write('input a2= ');read(a2);



write('input b2= ');read(b2); write('input c2= ');read(c2); d:=abs(c1-c2)/sqrt(a1*a1+b1*b1); writeln('d((d1),(d2))= ',d:8:12);readln; readln; End.

៨.២៣.កវរដនៃនឡណឡសពរបងបនផ តពរ

គឲរន តនែ ( )1 1 1 1: 0d a x b y c+ + = ន ( )2 2 2 2: 0d a x b y c+ + = ន គៗ ល

ចបែែែលមវធគណនលបអែេនចណចនែយន តយ នែលល var a1,b1,c1,a2,b2,c2,d,dx,dy,x,y:real; begin write('input a1= ');read(a1); write('input b1= ');read(b1); write('input c1= ');read(c1); write('input a2= ');read(a2); write('input b2= ');read(b2); write('input c2= ');read(c2); d:=a1*b2-a2*b1; dx:=b1*c2-b2*c1; dy:=a2*c1-a1*c2; x:=dx/d;y:=dy/d; writeln('x= ',x:8:12);readln; writeln('y= ' ,y:8:12);readln; End.

៨.២៤.កវរដនកពលតណនមកពរបនផ តបសព រ

គឲរន តមនមេែែម គៗ ( ) ( )1 1 1 1 2 2 2 2: 0 , : 0d a x b y c d a x b y c+ + = + + =

ន ( )3 3 3 3: 0d a x b y c+ + = ន គៗ តេនជតេណ ABC មដល

ចបែែែលមវធគណនលបអែេនៃនលនបលយ ែតេណល

កមវធគណនកវរដនកពលតណ

var a,b,c,a1,b1,c1,a2,b2,c2,a3,b3,c3,xa,ya,xb,yb,xc,yc,xg,yg,xo,yo,xh,yh,xi,yi,p,q,s,v,R,ro,da,db,dc,d:real; begin write('input a1=');read(a1); write('input b1=');read(b1); write('input c1=');read(c1); write('input a2=');read(a2); write('input b2=');read(b2); write('input c2=');read(c2); write('input a3=');read(a3); write('input b3=');read(b3); write('input c3=');read(c3); da:=a2*b3-a3*b2; dc:=a1*b3-a3*b1; db:=a1*b2-a2*b1;



xa:=(-c2*b3+c3*b2)/da; ya:=(-a2*c3+a3*c2)/da; xb:=(-c1*b2+c2*b1)/db; yb:=(-a1*c2+a2*c1)/db; xc:=(-c1*b3+c3*b1)/dc; yc:=(-a1*c3+a3*c1)/dc; writeln(' xa= ',xa:8:12, ' ya= ' , ya:8:12); readln; writeln(' xb= ',xb:8:12, ' yb= ' , yb:8:12); writeln(' xc= ',xc:8:12, ' yc= ' , yc:8:12); readln; End.

៨.២៥.សម ររងង រកតណែដលនមកពបនផ តបសព រ

គឲរន តមនមេែែម គៗ ( ) ( )1 1 1 1 2 2 2 2: 0 , : 0d a x b y c d a x b y c+ + = + + =

ន ( )3 3 3 3: 0d a x b y c+ + = ន គៗ តេនជតេណ ABC មដល

ចបែែែលមវធែលមេែៃនែ ាែ ចលកៃនតេណល var a1,b1,c1,a2,b2,c2,a3,b3,c3,xa,ya,xb,yb,xc,yc,xo,yo,p,q,u,v,R,R1,d12,d13,d23,dxa,dya,dxb,dyb,dxc,dyc:real; begin write('input a1=');read(a1); write('input b1=');read(b1); write('input c1=');read(c1); write('input a2=');read(a2); write('input b2=');read(b2); write('input c2=');read(c2); write('input a3=');read(a3); write('input b3=');read(b3); write('input c3=');read(c3); d23:=a2*b3-a3*b2; d13:=a1*b3-a3*b1; d12:=a1*b2-a2*b1; dxa:=c2*b3-c3*b2; dya:=a2*c3-a3*c2; dxb:=c1*b2-c2*b1; dyb:=a1*c2-a2*c1; dxc:=c1*b3-c3*b1; dyc:=b1*c3-b3*c1; xa:=dxa/d23; ya:=dya/d23; xb:=dxb/d12; yb:=dyb/d12; xc:=dxc/d13; yc:=dyc/d13; p:=(xc*xc-xa*xa)*(yb-ya)-(xb*xb-xa*xa)*(yc-ya)+(yb-ya)*(yc-yb)*(yc-ya); q:=(xc-xa)*(yb-ya)-(xb-xa)*(yc-ya); u:=(xb-xa)*(yc*yc-ya*ya)-(xc-xa)*(yb*yb-ya*ya)+(xb-xa)*(xc-xb)*(xc-xa); v:=(xb-xa)*(yc-ya)-(xc-xa)*(yb-ya);



xo:=p/(2*q); yo:=u/(2*v); R:=sqrt(sqr(xo-xa)+sqr(yo-ya)); R1:=R*R; writeln(' xo= ',xo:8:12, ' yo= ' , yo:8:12); readln; writeln('(x-xo)^2+(y-yo)^2=', R1:8:12); readln; End.

៨.២៦.កវរដនផផ តរងង ដយស ល កណបណពេនងបនផ តពរែដលសព រ

គឲរន តនែ ( )1 1 1 1: 0d a x b y c+ + = ន ( )2 2 2 2: 0d a x b y c+ + = ន គៗ ល

ចបែែែលមវធែលលបអែេនផាតែ យ នងនលមនេR ដ ហនទនចន តយ នែល var a1,b1,c1,a2,b2,c2,R,x1,y1,x2,y2,x3,y3,x4,y4,d,dx1,dy1,dx2,dy2,dx3,dy3,dx4,dy4,u,v:real; begin write('input a1=');read(a1); write('input b1=');read(b1); write('input c1=');read(c1); write('input a2=');read(a2); write('input b2=');read(b2); write('input c2=');read(c2); write('input R=');read(R); a1:=3;b1:=4;c1:=5;a2:=4;b2:=3;c2:=6; R:=6; u:=R*sqrt(a1*a1+b1*b1); v:=R*sqrt(a2*a2+b2*b2); d:=a1*b2-a2*b1; dx1:=(-c1+u)*b2-(-c2+v)*b1; dy1:=a1*(-c2+v)-a2*(-c1+u); x1:=dx1/d; y1:=dy1/d; dx2:=(-c1-u)*b2-(-c2-v)*b1; dy2:=a1*(-c2-v)-a2*(-c1-u); x2:=dx2/d; y2:=dy2/d; dx3:=(-c1-u)*b2-(-c2+v)*b1; dy3:=a1*(-c2+v)-a2*(-c1-u); x3:=dx3/d; y3:=dy3/d; dx4:=(-c1+u)*b2-(-c2-v)*b1; dy4:=a1*(-c2-v)-a2*(-c1+u); x4:=dx4/d; y4:=dy4/d; writeln(' x1= ',x1:8:12, ' y1= ' , y1:8:12); readln; writeln(' x2= ',x2:8:12, ' y2= ' , y2:8:12); writeln(' x3= ',x3:8:12, ' y3= ' , y3:8:12); writeln(' x4= ',x4:8:12, ' y4= ' , y4:8:12);readln; End.



៩.ធែណមាណកៗក ាល

៩.១.កវរដនឡណឡកក លៃនវងអតម

គឲរនែចណច ( ), ,a a aA x y z ន ( ), ,b b bB x y z ល

ចបែែែលមវធគណនលបអែេនចណច M លចយ លអ ត AB ល

ែបមនយ , ,2 2 2

a b a b a bm m m

x x y y z zx y z+ + += = =

កមវធ var xa,ya,za,xb,yb,zb,xm,ym,zm:real; begin write('input xa=' );read(xa); write('input ya= ' ); read(ya); write('input za= ' );read(za); write('input xb = ' ) ;read(xb); write('input yb= ' ); read(yb); write('input zb= ' );read(zb); xm:=(xa+xb)/2; ym:=(ya+yb)/2; zm:=(za+zb)/2; writeln('xm= ' ,xm:8:12);readln; writeln('ym= ',ym:8:12); writeln('zm= ',zm);readln; End.

៩.២.ឡមទ របងពរឡណឡករងលក


ចបែែែលមវធគណនចមរ ដែយនែចណច A ន B ល

ែបមនយ ( ) ( ) ( ) ( )2 2 2b a b a b ad AB x x y y z z= − + − + −

កមវធគណន var xa,ya,za,xb,yb,zb,d:real; begin write('input xa=');read(xa); write('input ya= ' ); read(ya); write('input za= ' );read(za); write('input xb = ' ) ;read(xb); write('input yb= ' ); read(yb); write('input zb= ' );read(zb); d:=sqrt(sqr(xb-xa)+sqr(yb-ya)+sqr(zb-za)); writeln('d(AB)= ' ,d:8:12);readln; readln; End.



៩.៣.កវរដនទ ទមទនត ណករងលក

គឲរចណច ( ) ( ), , , , ,a a a b b bA x y z B x y z ន ( ), ,c c cC x y z តេនតេណមដល

ចបែែែលមវធគណនលបអែេនលជលមរនៃនតេណ ABC ល var xa,ya,za,xb,yb,zb,xc,yc,zc,xg,yg,zg:real; begin write('input xa=' );read(xa); write('input ya= ' ); read(ya); write('input za= ' );read(za); write('input xb = ' ) ;read(xb); write('input yb= ' ); read(yb); write('input zb= ' );read(zb); write('input xc = ' ) ;read(xc); write('input yc= ' ); read(yc); write('input zc= ' );read(zc); xg:=(xa+xb+xc)/3; yg:=(ya+yb+yc)/3; zg:=(za+zb+zc)/3; writeln('xg= ' ,xg:8:12);readln; writeln('yg= ',yg:8:12); writeln('zg= ',zg);readln; End.

៩.៤.កវរដនឡណឡែឡកករងៃនវងអតមករងលក


ចបែែែលមវធគណនលបអែេនចណច P ងចលលគអ ត AB តមផលធម PA kPB

= ល

ែបមនយ , ,1 1 1a b a b a b

p p px kx y ky z kzx y z

k k k+ + +

= = =+ + +

កមវធគណន var xa,ya,za,xb,yb,zb,xp,yp,zp,k:real; begin write('input xa=' );read(xa); write('input ya= ' ); read(ya); write('input za= ' );read(za); write('input xb = ' ) ;read(xb); write('input yb= ' ); read(yb); write('input zb= ' );read(zb); write('input k = ' ) ;read(k); xp:=(xa+k*xb)/(1+k); yp:=(ya+k*yb)/(1+k); zp:=(za+k*xb); writeln('xp= ' ,xp:8:12);readln; writeln('yp= ' ,yp:8:12); writeln('zp= ' ,zp);readln; End.



៩.៥.កវរដនឡណឡែឡកៃនវងអតមករងលក


ចបែែែលមវធគណនលបអែេនចណចQ ងចលកអ ត AB តមផលធម QA kQB

= ល

ែបមនយ , ,1 1 1a b a b a b

p p px kx y ky z kzx y z

k k k− − −

= = =− − −


var xa,ya,za,xb,yb,zb,xq,yq,zq,k:real; begin write('input xa=' );read(xa); write('input ya= ' ); read(ya); write('input za= ' );read(za); write('input xb = ' ) ;read(xb); write('input yb= ' ); read(yb); write('input zb= ' );read(zb); write('input k = ' ) ;read(k); xq:=(xa+k*xb)/(1+k); yq:=(ya+k*yb)/(1+k); zq:=(za+k*xb); writeln('xq= ' ,xq:8:12);readln; writeln('yq= ' ,yq:8:12); writeln('zq= ' ,zq);readln; End.

៩.៦.ផលគណយអ ែលរបងពរវឡទចរករងលក

គឲរវចលែនែ ( ), ,u u uu x y z→

= ន ( ), ,v v vv x y z→

= ល

ចបែែែលមវធគណនផលគណណ ងលu v→ →

⋅ ល

របមន u v u v u vu v x x y y z z→ →

⋅ = + + ។


var xu,yu,zu,xv,yv,zv,u,v,uv:real; begin write('input xu=' );read(xu); write('input yu= ' ); read(yu); write('input zu= ' );read(zu); write('input xv = ' ) ;read(xv); write('input yv= ' ); read(yv); write('input zv= ' );read(zv); uv:=xu*xv+yu*yv+zu*zv; writeln('u.v= ' ,uv:8:12);readln; End.



៩.៧.កសនសៃនមរបងពរវឡទចរ

គឲរវចលែនែ ( ), ,u u uu x y z→

= ន ( ), ,v v vv x y z→

= ល

ចបែែែលមវធគណនលបនបៃនមែយនែវចលែ &u v→ →

ល

ែបមនយ 2 2 2 2 2 2

cos u v u v u v

u u u v v v

x x y y z zwx y z x y z

+ +=

+ + + + ល

កមវធគណន var xu,yu,zu,xv,yv,zv,u,v,cosw:real; begin write('input xu=' );read(xu); write('input yu= ' ); read(yu); write('input zu= ' );read(zu); write('input xv = ' ) ;read(xv); write('input yv= ' ); read(yv); write('input zv= ' );read(zv); cosw:=(xu*xv+yu*yv+zu*zv)/(sqrt(sqr(xu)+sqr(yu)+sqr(zu))*sqrt(sqr(xv)+sqr(yv)+sqr(zv))); writeln('cosw= ' ,cosw:8:12);readln; readln; End.

៩.៨.ែវងងតណករងលក


ចបែែែលមវធគណនងវជរយ ៃនតេណ ABC ល


var xa,ya,za,xb,yb,zb,xc,yc,zc,a,b,c:real; begin write('input xa=' );read(xa); write('input ya= ' ); read(ya); write('input za= ' );read(za); write('input xb = ' ) ;read(xb); write('input yb= ' ); read(yb); write('input zb= ' );read(zb); write('input xc = ' ) ;read(xc); write('input yc= ' ); read(yc); write('input zc= ' );read(zc); a:=sqrt(sqr(xc-xb)+sqr(yc-yb)+sqr(zc-zb)); b:=sqrt(sqr(xc-xa)+sqr(yc-ya)+sqr(zc-za)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)+sqr(zb-za)); writeln('BC=a= ' ,a:8:12);readln; writeln('AC=b= ' ,b:8:12); writeln('AB=c= ' ,c:8:12);readln; End.



៩.៩.កវរដនផផ តរងង រកករងតណករងលក


ចបែែែលមវធគណនលបអែេនផាតែ ាែ ចលលគៃនតេណ ABC ល

របមន 0 0 0, ,a b c a b c a b cax bx cx ay by cy az bz czx y za b c a b c a b c+ + + + + +

= = =+ + + + + +

កមវធ

var xa,ya,za,xb,yb,zb,xc,yc,zc,a,b,c,xo,yo,zo:real;

begin

write('input xa=' );read(xa);

write('input ya= ' ); read(ya);

write('input za= ' );read(za);

write('input xb = ' ) ;read(xb);

write('input yb= ' ); read(yb);

write('input zb= ' );read(zb);

write('input xc = ' ) ;read(xc);

write('input yc= ' ); read(yc);

write('input zc= ' );read(zc);

a:=sqrt(sqr(xc-xb)+sqr(yc-yb)+sqr(zc-zb));

b:=sqrt(sqr(xc-xa)+sqr(yc-ya)+sqr(zc-za));

c:=sqrt(sqr(xb-xa)+sqr(yb-ya)+sqr(zb-za));

xo:=(a*xa+b*xb+c*xc)/(a+b+c);

yo:=(a*ya+b*yb+c*yc)/(a+b+c);

zo:=(a*za+b*zb+c*zc)/(a+b+c);

writeln('xo= ' ,xo:8:12);readln;

writeln('yo= ' ,yo:8:12);

writeln('zo= ' ,zo:8:12);readln;

End.

៩.១០.ៃផផករនងរងង រកតណករងលក


ចបែែែលមវធគណនៃផលកនេែ ាែ ចលកៃនតេណ ABC ល




var xa,ya,za,xb,yb,zb,xc,yc,zc,a,b,c,p,R,s:real;

begin

write('input xa=' );read(xa);

write('input ya= ' ); read(ya);

write('input za= ' );read(za);

write('input xb = ' ) ;read(xb);

write('input yb= ' ); read(yb);

write('input zb= ' );read(zb);

write('input xc = ' ) ;read(xc);

write('input yc= ' ); read(yc);

write('input zc= ' );read(zc);

a:=sqrt(sqr(xc-xb)+sqr(yc-yb)+sqr(zc-zb));

b:=sqrt(sqr(xc-xa)+sqr(yc-ya)+sqr(zc-za));

c:=sqrt(sqr(xb-xa)+sqr(yb-ya)+sqr(zb-za)); p:=(a+b+c)/2; s:=sqrt(p*(p-a)*(p-b)*(p-c)); R:=(a*b*c)/(4*s); writeln('s= ' ,s:8:12);readln; writeln('R= ' ,R:8:12);readln; End.

៩.១១.ៃផផករនងរងង រកករងតណករងលក


ចបែែែលមវធគណនៃផលកនេែ ាែ ចលលគៃនតេណ ABC ល


var xa,ya,za,xb,yb,zb,xc,yc,zc,a,b,c,p,r,s:real; begin write('input xa=' );read(xa); write('input ya= ' ); read(ya); write('input za= ' );read(za); write('input xb = ' ) ;read(xb); write('input yb= ' ); read(yb); write('input zb= ' );read(zb); write('input xc = ' ) ;read(xc); write('input yc= ' ); read(yc); write('input zc= ' );read(zc); a:=sqrt(sqr(xc-xb)+sqr(yc-yb)+sqr(zc-zb)); b:=sqrt(sqr(xc-xa)+sqr(yc-ya)+sqr(zc-za)); c:=sqrt(sqr(xb-xa)+sqr(yb-ya)+sqr(zb-za)); p:=(a+b+c)/2; s:=sqrt(p*(p-a)*(p-b)*(p-c));



r:=s/p; writeln('s= ' ,s:8:12);readln; writeln('r= ' ,r:8:12);readln; End.

៩.១២.ផលគណរបងពរវឡទចករងលក

គឲរវចលែ ( ), ,u u uu x y z→

= ន ( ), ,v v vv x y z→

=

ចបែែែលមវធគណនផលគណវចលែ u v→ →

× ល

របមន ( ), ,uv uv uvu v x y z→ →

× = ងនល uv u v v u

uv v u u v

uv u v v u

x y z y zy x z x zz x y x y

= − = − = −


var xu,yu,zu,xv,yv,zv,xuv,yuv,zuv:real; begin write('input xu=' );read(xu); write('input yu= ' ); read(yu); write('input zu= ' );read(zu); write('input xv = ' ) ;read(xv); write('input yv= ' ); read(yv); write('input zv= ' );read(zv); xuv:=yu*zv-yv*zu; yuv:=xv*zu-xu*zv; zuv:=xu*yv-xv*yu; writeln('xuv= ',xuv:8:12 );readln; writeln('yuv= ',yuv:8:12 ); writeln('zuv= ',zuv:8:12 );readln; readln; End.

៩.១៣.ផលគណឡមព

គឲរវចលែ ( ) ( ), , , , ,u u u v v vu x y z v x y z→ →

= = ន ( ), ,w w ww x y z→

=

ចបែែែលមវធគណនផលគណចមរន ( ).u v w→ → →

× ល

របមន ា

( ) ( )( ). ( )u u u

v v v w u v v u w u v v u w u v v u

w w w

x y zu v w x y z x y z y z y x z x z z x y x y

x y z

→ → →

× = = − − − + −

កមវធគណន var xu,yu,zu,xv,yv,zv,xw,yw,zw,xuv,yuv,zuv,produitmix:real; begin write('input xu=' );read(xu); write('input yu= ' ); read(yu);



write('input zu= ' );read(zu);

write('input xv = ' ) ;read(xv);

write('input yv= ' ); read(yv);

write('input zv= ' );read(zv);

xuv:=yu*zv-yv*zu;

yuv:=xv*zu-xu*zv;

zuv:=xu*yv-xv*yu;

produitmix:=xw*xuv+yw*yuv+zw*zuv;

writeln('produitmix= ',produitmix:8:12 );readln;

readln;

End. ៩.១៤.ផលគណបវឡទចរករងលក

គឲរវចលែ ( ) ( ), , , , ,u u u v v vu x y z v x y z→ →

= = ន ( ), ,w w ww x y z→

=

ចបែែែលមវធគណនផលគណវចលែ ( )u v w→ → →

× × ល

របមន ( ) ( . ) ( . )u v w u w v u v w→ → → → → → → → →

× × = −

កមវធ var xu,yu,zu,xv,yv,zv,xw,yw,zw,p,q,xt,yt,zt:real; begin write('input xu=' );read(xu); write('input yu= ' ); read(yu); write('input zu= ' );read(zu); write('input xv = ' ) ;read(xv); write('input yv= ' ); read(yv); write('input zv= ' );read(zv); p:=xu*xw+yu*yw+zu*zw; q:=xu*xv+yu*yv+zu*zv; xt:=p*xv-q*xw; yt:=p*yv-q*yw; zt:=p*zv-q*zw; writeln('xt=',xt:8:12 );readln; writeln('yt=',yt:8:12); writeln('zt=' ,zt:8:12);readln; End.

៩.១៥.ផលគណឡមពបនវឡទចរករងលក

គឲរវចលែ ( ) ( ) ( ), , , , , , , ,x y z x y z x y za a a a b b b b c c c c→ → →

= = = ន ( ), ,x y zd d d d→

=

ចបែែែលមវធគណន ( ).( )a b c d→ → → →

× × ល



របមន . .( ).( ). .

a c a da b c db c b d

→ → → →→ → → →

→ → → →× × =

កមវធ var ax,ay,az,bx,by,bz,cx,cy,cz,dx,dy,dz,p1,p2,p3,p4,p:real; begin write('input ax= ' );read(ax); write('input ay= ' );read(ay); write('input az= ' );read(az); write('input bx = ' );read(bx); write('input by= ' ); read(by); write('input bz= ' );read(bz); write('input cx= ' );read(cx); write('input cy= ' );read(cy); write('input cz= ' );read(cz); write('input dx = ' );read(dx); write('input dy= ' ); read(dy); write('input dz= ' );read(dz); p1:=ax*cx+ay*cy+az*cz; p2:=ax*dx+ay*dy+az*dz; p3:=bx*cx+by*cy+bz*cz; p4:=bx*dx+by*dy+bz*dz; p:=p1*p4-p3*p2; writeln('(axb).(cxd)=',p:8:12 );readln; End.

៩.១៦.ៃផផករតណករងលក

គឲរចណច ( ) ( ) ( ), , ; , , ; , ,a a a b b b c c cA x y z B x y z C x y z តេនជតេណមដល

ចបែែែលមវធគណនៃផលកៃនតេណ ABC ល

របមន 12ABCS AB AC

→ →

= × ។

កមវធគណន var xa,ya,za,xb,yb,zb,xc,yc,zc,s,u,v,w:real; begin write('input xa= ' );read(xa); write('input ya= ' );read(ya); write('input za= ' );read(za); write('input xb = ' );read(xb); write('input yb= ' ); read(yb); write('input zb= ' );read(zb); write('input xc= ' );read(xc); write('input yc= ' );read(yc); write('input zc= ' );read(zc); u:=(yb-ya)*(zc-za)-(yc-ya)*(zb-za); v:=(xc-xa)*(zb-za)-(xb-xa)*(zc-za); w:=(xb-xa)*(yc-ya)-(xc-xa)*(yb-ya); s:=(sqrt(sqr(u)+sqr(v)+sqr(w)))/2; writeln('s=',s:8:12 );readln; readln; End.

A

B C



៩.១៧.មឡតមខខតែវត

គឲរចណច ( ) ( ) ( ), , ; , , ; , ,a a a b b b c c cA x y z B x y z C x y z ន ( ), ,d d dD x y z

តេនជចតមមដល ចបែែែលមវធគណនមឌែចតម ABCD ល

របមន 1 ( ).ABCDV AB AC ADc

→ → →

= × ។


var xa,ya,za,xb,yb,zb,xc,yc,zc,xd,yd,zd,V,p,q,r,s:real; begin write('input xa= ' );read(xa); write('input ya= ' );read(ya); write('input za= ' );read(za); write('input xb = ' );read(xb); write('input yb= ' ); read(yb); write('input zb= ' );read(zb); write('input xc= ' );read(xc); write('input yc= ' );read(yc); write('input zc= ' );read(zc); p:=(yb-ya)*(zc-za)-(yc-ya)*(zb-za); q:=(xc-xa)*(zb-za)-(xb-xa)*(zc-za); r:=(xb-xa)*(yc-ya)-(xc-xa)*(yb-ya); s:=p*(xd-xa)+q*(yd-ya)+r*(zd-za); V:=(abs(s))/6; writeln('V=',v:8:12 );readln; readln; End.

A

B

C

D



៩.១៨.ឡមទ ពឡណឡមេមង ករងលក

var xa,ya,za,a,b,c,d,p,q,dap:real; begin write('input xa= ' );read(xa); write('input ya= ' );read(ya); write('input za= ' );read(za); write('input a = ' );read(a); write('input b= ' ); read(b); write('input c= ' );read(c); write('input d= ' );read(d); p:=abs(a*xa+b*ya+c*za+d); q:=sqrt(a*a+b*b+c*c); dap:=p/q; writeln('d(A,(p))=',dap:8:12 );readln; readln; End.

៩.១៩.ឡមទ ពឡណឡមេបនផ តករងលក

គឲរន ត ( )L េតតម ( )0 0 0, ,M x y z ដន ( ), ,u a b c→

= ល

ចបែគណនចមរ ដនចណច ( ), ,a a aA x y z ទន ត ( )L ល

របមន ( ) | |,( )| |

AM ud A Lu

→ →

→

×= ។


var xa,ya,za,a,b,c,xo,yo,zo,u,v,w,p,q,dal:real; begin write('input xa= ' );read(xa); write('input ya= ' );read(ya); write('input za= ' );read(za); write('input xo= ' );read(xo); write('input yo= ' );read(yo); write('input zo= ' );read(zo); write('input a = ' );read(a); write('input b= ' ); read(b); write('input c= ' );read(c); u:=c*(yo-ya)-b*(zo-za); v:=a*(zo-za)-c*(xo-xa); w:=b*(xo-xa)-a*(yo-ya); p:=sqrt(u*u+v*v+w*w); q:=sqrt(a*a+b*b+c*c); dal:=p/q; writeln('d(A,(L))=',dal:8:12 );readln; End.

( )L

A

M

H( ),d A L

•

u→



៩.២០.ឡមទ របងមង ពរសប រ

គឲរ ថនែ គៗ ( )1 1 1 1 1: 0p a x b y c z d+ + + = ន ( )2 2 2 2 2: 0p a x b y c z d+ + + =

ងនល 1 1 1

2 2 2

a b ca b c

= = ល ចបែែែលមវធគណនចមរ ដែយ ថយ នែល

របមន ( ) 2 11 2 2 2 2

1 1 1

| |, d dd p pa b c

−=

+ +

កមវធ

var a1,b1,c1,d1,a2,b2,c2,d2,dp1p2:real; begin write('input a1= ' );read(a1); write('input b1= ' );read(b1); write('input c1= ' );read(c1); write('input d1= ' );read(d1); write('input a2= ' );read(a2); write('input b2= ' );read(b2); write('input c2= ' );read(c2); write('input d2= ' );read(d2); dp1p2:=(abs(d2-d1))/(sqrt(a1*a1+b1*b1+c1*c1)); writeln('d((p1),(p2))=',dp1p2:8:12 );readln; readln; End.

៩.២១.ឡមទ របងបនផ តពរករងលក

គឲរន ត ( )1L េតតម ( )1 1 1 1, ,M x y z នវចលែ ( )1 1 1 1, ,u a b c→

=

ន ត ( )2L េតតម ( )2 2 2 2, ,M x y z នវចលែ ( )2 2 2 2, ,u a b c→

= ល

ចបែែែលមវធគណនចមរ ដែយន ត ( )1L ន( )2L ល

របមន ( )1 2 1 2

1 2

1 2

.,

| |

u u M Md L L

u u

→ → →

→ →

× =

× ។


var x1,y1,z1,a1,b1,c1,x2,y2,z2,a2,b2,c2,p,q,u,v,w,dl1l2:real; begin write('input x1= ' );read(x1); write('input y1= ' );read(y1); write('input z1= ' );read(z1); write('input a1= ' );read(a1); write('input b1= ' );read(b1); write('input c1= ' );read(c1); write('input x2= ' );read(x2); write('input y2= ' );read(y2);



write('input z2= ' );read(z2); write('input a2= ' );read(a2); write('input b2= ' );read(b2); write('input c2= ' );read(c2); u:=b1*c2-b2*c1; v:=a2*c1-a1*c2; w:=a1*b2-a2*b1; p:=abs(u*(x2-x1)+v*(y2-y1)+w*(z2-z1)); q:=sqrt(u*u+v*v+w*w); dl1l2:=p/q; writeln('d((L1),(L2))=',dl1l2:8:12 );readln; readln; End.

៩.២២.កវរដនឡណឡសពរបងបនផ តនងមង

គឲរន ត ( )L េតតម ( )0 0 0, ,M x y z នវចលែ ( ), ,u a b c→

=

ន ថ ( ) : 0p u x v y w z r+ + + = ល

ចបែែែលមវធគណនលបអែេនចណចនែយន ត ( )L ជមដ ថ ( )p ល

ដកពយ

មេែេហ ាហ ងមហតៃន ( )L េតតម ( )0 0 0, ,M x y z នវចលែ ( ), ,u a b c→

= ែែរ

( )0

0

0

:x x at

L y y btz z ct

= + = + = +

ដលមេែ ( )L ជន លគ ( )p គេន 0 0 0( ) ( ) ( ) 0u x at v y bt w z ct r+ + + + + + =

គយញ 0 0 00

ux vy wz rt tau bv cw+ + +

= − =+ +

ល

ដលតៃមថt ជន លគ ( )L គេនលបអែេនចណចន P ែយន ត ( )L ន ថ ( )p គគរ

( )0 0 0 0 0 0, ,P x at y bt z ct+ + + ល


var a,b,c,u,v,w,r,xo,yo,zo,t,xp,yp,zp:real; begin write('a=');read(a); write('b=');read(b); write('c=');read(c); write('xo=');read(xo); write('yo=');read(yo); write('zo=');read(zo); write('u=');read(u); write('v=');read(v); write('w=');read(w);



write('r=');read(r); t:=-(u*xo+v*yo+w*zo+r)/(a*u+b*v+c*w); xp:=xo+a*t; yp:=yo+b*t; zp:=zo+c*t; writeln('xp=' ,xp:8:12);readln; writeln('yp=' ,yp:8:12); writeln('zp=' ,zp:8:12);readln; End.

៩.២៣.កវរដនឡណឡជ ណ ងៃនឡកលែកងរបសឡណឡមេលណមង ករងលក

គឲរចណច ( ), ,a a aA x y z ន ថ ( ) : 0p ax by cz d+ + + = ល

ចបែែែលមវធគណនលបអែេនៃនចណច H ជជៃនចចលងលៃនចណច A ល ថ( )p ល

ដកពយ

ដល ( ) ( ), ,h h hH x y z p∈ នន ( )0 1h h hax by cz d+ + + =

គមន ( ), ,h a h a h aAH x x y y z z→

= − − −

េដ ( )AH p→

⊥ នន / / pAH n→ →

មមបល pAH t n→ →

= ងនល ( ), ,pn a b c→

=

គេន ( )2h a

h a

h a

x x aty y btz z ct

= + = + = +

ដល( )2 ជ លគ ( )1 នន ( ) ( ) ( ) 0a a aa x at b y bt c z ct d+ + + + + + =

គយញ 02 2 2a a aax by cz dt ta b c+ + +

= − =+ +

ដលជ លគ ( )2 គេនរ

( )0 0 0, ,a a aH x at y bt z ct+ + + ល


var a,b,c,d,t,xa,ya,za,xh,yh,zh:real;

begin

write('a=');read(a);

write('b=');read(b);

write('c=');read(c);

write('d=');read(d);

write('xa=');read(xa);

write('ya=');read(ya);

write('za=');read(za);

t:=-(a*xa+b*ya+c*za+d)/(a*a+b*b+c*c);

xh:=xa+a*t;

A

H

p•



yh:=ya+b*t;

zh:=za+c*t;

writeln('xh=' ,xh:8:12);readln;

writeln('yh=' ,yh:8:12);

writeln('zh=' ,zh:8:12);readln;

End.

៩.២៤.កវរដនឡណឡឆមព រ ធៀបនងមង មករងលក


ចបែែែលមវធគណនលបអែេនៃនចណច B ជចណចឆថន គៗ ៃនចណច A ធមនច ថ ( )p ល

របៀបយ

ដលH ជជៃនចចលងលៃនចណច A ល ថ( )p ល

B ជចណចឆថន គៗ ៃនចណច A ធម

ន ថ ( )p នន H ជចណចលចយ លៃន AB ល

តតវែលលបអែេនៃនH រមលែម ៩.២៣)ល លបអែេនៃនចណច B គគ 2 , 2 , 2b h a b h a b h ax x x y y y z z z= − = − = − ល

កមវធដពយ

var a,b,c,d,t,xa,ya,za,xh,yh,zh,xb,yb,zb:real; begin write('a=');read(a); write('b=');read(b); write('c=');read(c); write('d=');read(d); write('xa=');read(xa); write('ya=');read(ya); write('za=');read(za); t:=-(a*xa+b*ya+c*za+d)/(a*a+b*b+c*c); xh:=xa+a*t;

A

B

Hp



yh:=ya+b*t; zh:=za+c*t; xb:=2*xh-xa; yb:=2*yh-ya; zb:=2*zh-za; writeln('xb=' ,xb:8:12);readln; writeln('yb=' ,yb:8:12); writeln('zb=' ,zb:8:12);readln; End.

៩.២៥.សម រែសបណពនងមង មករងលកនងកវរដនៃនឡណឡបណព


ចបែែែលមវធែលមេែយេៃនងស ( )S មនផាត A ដ ហននច ថ ( )p ែចលណតលបអែេនៃន

ចណច ហន H ែយងស ( )S ន ថ ( )p ល

កមវធដពយ

var a,b,c,d,t,xa,ya,za,R,xh,yh,zh:real; begin write('a=');read(a); write('b=');read(b); write('c=');read(c); write('d=');read(d); write('xa=');read(xa); write('ya=');read(ya); write('za=');read(za); R:=(abs(a*xa+b*ya+c*za+d))/sqrt(a*a+b*b+c*c); t:=-(a*xa+b*ya+c*za+d)/(a*a+b*b+c*c); xh:=xa+a*t; yh:=ya+b*t; zh:=za+c*t; writeln('(x-xa)^2+(y-ya)^2+(z-za)^2=',R:8:12);readln; writeln('xh=' ,xh:8:12); writeln('yh=' ,yh:8:12); writeln('zh=' ,zh:8:12);readln; End.

H

A

R

p

( )S



៩.២៦.កវរដនៃនផផ តរបសែសបណពនងមង តង ឡណឡម


ចបែែែលមវធែលលបអែេនៃនផាតែងស ( )S មនេR ដ ហននច ថ ( )p ត A ល

របៀបដពយ

ដល ( ), ,i i iI x y z ជផាតែងស ( )S

គេន ( ), ,i a i a i aAI x x y y z z→

= − − −

េដ / / pAI n→ →

នន i a

i a

i a

x x aty y btz z ct

= + = + = +

មដហ លមត pAI tn→ →

= នន 02 2 2

| |

| |p

AI Rt ta b cn

→

→= ± = ± = ±+ +

លបអែេនផាតយ នែគគ រ

( )1 0 0 0, ,a a aI x at y bt z ct+ + + ន ( )2 0 0 0, ,a a aI x at y bt z ct− − − ល


var a,b,c,d,xa,ya,za,R,t,x1,y1,z1,x2,y2,z2:real; begin write('a=');read(a); write('b=');read(b); write('c=');read(c); write('d=');read(d); write('xa=');read(xa); write('ya=');read(ya); write('za=');read(za); write('R=');read(R); t:=R/sqrt(a*a+b*b+c*c); x1:=xa+a*t; y1:=ya+b*t; z1:=za+c*t; x2:=xa-a*t; y2:=ya-b*t; z2:=za-c*t; writeln('x1=' ,x1:8:12);readln; writeln('y1=' ,y1:8:12); writeln('z1=' ,z1:8:12); writeln('x2=' ,x2:8:12); writeln('y2=' ,y2:8:12); writeln('z2=' ,z2:8:12);readln; End.



៩.២៧.កវរដនៃនផផ តនងែវងរបសែសរកឡតមខម

គឲរចណច ( ) ( ) ( ), , , , , , , ,a a a b b b c c cA x y z B x y z C x y z ន ( ), ,d d dD x y z តេន

ជចតមមដ ល

ចបែែែលមវធគណនលបអែេនផាតនែង េែងសាែ ចលកចតម ABCD ល var xa,ya,za,xb,yb,zb,xc,yc,zc,xd,yd,zd,sa,sb,sc,sd,a,b,c,d,ds,da,db,dc,r,xi,yi,zi,m,n,p,q:real; begin write('input xa= ');read(xa); write('input ya= ');read(ya); write('input za= ');read(za); write('input xb= ');read(xb); write('input yb= ');read(yb); write('input zb= ');read(zb); write('input xc= ');read(xc); write('input yc= ');read(yc); write('input zc= ');read(zc); write('input xd= ');read(xd); write('input yd= ');read(yd); write('input zd= ');read(zd); sa:=xa*xa+ya*ya+za*za; sb:=xb*xb+yb*yb+zb*zb; sc:=xc*xc+yc*yc+zc*zc; sd:=xd*xd+yd*yd+zd*zd; m:=(za-zb)*((xa-xc)*(ya-yd)-(xa-xd)*(ya-yc)); n:=(za-zb)*((sc-sa)*(ya-yd)-(sd-sa)*(ya-yc)); p:=(za-zb)*((xa-xc)*(sd-sa)-(xa-xd)*(sc-sa)); q:=(sb-sa)*((xa-xc)*(ya-yd)-(xa-xd)*(ya-yc)); ds:=(xa-xb)*((ya-yc)*(za-zd)-(ya-yd)*(za-zc))-(ya-yb)*((xa-xc)*(za-zd)-(xa-xd)*(za-zc))+m; da:=(sb-sa)*((ya-yc)*(za-zd)-(ya-yd)*(za-zc))-(ya-yb)*((sc-sa)*(za-zd)-(sd-sa)*(za-zc))+n; db:=(xa-xb)*((sc-sa)*(za-zd)-(sd-sa)*(za-zc))-(sb-sa)*((xa-xc)*(za-zd)-(xa-xd)*(za-zc))+p; dc:=(xa-xb)*((ya-yc)*(sd-sa)-(ya-yd)*(sc-sa))-(ya-yb)*((xa-xc)*(sd-sa)-(xa-xd)*(sc-sa))+q; a:=da/ds; b:=db/ds; c:=dc/ds; d:=-sa-a*xa-b*ya-c*za; xi:=-a/2; yi:=-b/2; zi:=-c/2; R:=sqrt(sqr(xa-xi)+sqr(ya-yi)+sqr(za-zi)); writeln('xi= ',xi:8:12);readln; writeln('yi= ',yi:8:12); writeln('zi= ',zi:8:12); writeln('R= ',R:8:12); readln; End.



៩.២៨.កវរដនវរតសង នងផផ តរងង រកតណករងលក

គឲរចណច ( ) ( ) ( ), , , , , , , ,a a a b b b c c cA x y z B x y z C x y z តេនជតេណ ABC ល

គគបលមម ,AK BL ន CM ល ត ( ), ,h h hH x y z ជអែតបន ( )0 0 0, ,x y zΩ

ជផាតែ ាែ ចលកៃនតេណ ABC ល

ចបែែែលមវធគណនលបអែេនៃនចណច , , ,K L M H ន Ω ល

កមវធគណនរ var xa,ya,za,xb,yb,zb,xc,yc,zc,xh,yh,zh,xk,yk,zk,xl,yl,zl,xm,ym,zm,a1,a2,a3,b1,b2,b3,c1,c2,c3,t1,t2,t3,a,b,c,tk,d,n,u,v,w:real;

begin

write('input xa= ');read(xa);

write('input ya= ');read(ya);

write('input za= ');read(za);

write('input xb= ');read(xb);

write('input yb= ');read(yb);

write('input zb= ');read(zb);

write('input xc= ');read(xc);

write('input yc= ');read(yc);

write('input zc= ');read(zc);

c1:=xb-xa;

c2:=yb-ya;

c3:=zb-za;

b1:=xc-xa;

b2:=yc-ya;

b3:=zc-za;

a1:=xc-xb;

a2:=yc-yb;

a3:=zc-zb;

c:=sqr(c1)+sqr(c2)+sqr(c3);

a:=sqr(a1)+sqr(a2)+sqr(a3);

b:=sqr(b1)+sqr(b2)+sqr(b3);

t1:=-(a1*c1+a2*c2+a3*c3)/a;

t2:=-(a1*b1+a2*b2+a3*b3)/b;

t3:=(c1*b1+c2*b2+c3*b3)/c;

d:=-(c1+a1*t1)*(a2+b2*t2)+(a1+b1*t2)*(c2+a2*t1);



n:=-c1*(a2+b2*t2)+c2*(a1+b1*t2);

tk:=n/d;

xh:=xa+(c1+a1*t1)*tk;

yh:=ya+(c2+a2*t1)*tk;

zh:=za+(c3+a3*t1)*tk;

u:=(xa+xb+xc-xh)/2;

v:=(ya+yb+yc-yh)/2;

w:=(za+zb+zc-zh)/2;

xk:=xb+a1*t1;

yk:=yb+a2*t1;

zk:=zb+a3*t1;

xl:=xc+b1*t2;

yl:=yc+b2*t2;

zl:=zc+b3*t2;

xm:=xa+c1*t3;

ym:=ya+c2*t3;

zm:=za+c3*t3;

writeln('xk= ',xk:8:12);readln;

writeln('yk= ',yk:8:12);

writeln('zk= ',zk:8:12);

writeln('xl= ',xl:8:12);

writeln('yl= ',yl:8:12);

writeln('zl= ',zl:8:12);

writeln('xm= ',xm:8:12);

writeln('ym= ',ym:8:12);

writeln('zm= ',zm:8:12);

writeln('xh= ',xh:8:12);

writeln('yh= ',yh:8:12);

writeln('zh= ',zh:8:12);

writeln('xo= ',u:8:12);

writeln('yo= ',v:8:12);

writeln('zo= ',w:8:12);

readln;

End.



៩.២៩.កវរដនឡណឡសពរបងមង ប ា

គឲរ ថ ( ) ( )1 1 1 1 1 2 2 2 2 2: , : 0p a x b y c z d p a x b y c z d+ + + + + + =

ន ( )3 3 3 3 3: 0p a x b y c z d+ + + = ល

ចបែែែលមវធគណនលបអែេនៃនចណចន A ែយ ថយ ល ល


var a1,b1,c1,d1,a2,b2,c2,d2,a3,b3,c3,d3,d,dx,dy,dz,x,y,z:real; begin write('input a1=');read(a1); write('input b1=');read(b1); write('input c1=');read(c1); write('input d1=');read(d1); write('input a2=');read(a2); write('input b2=');read(b2); write('input c2=');read(c2); write('input d2=');read(d2); write('input a3=');read(a3); write('input b3=');read(b3); write('input c3=');read(c3); write('input d3=');read(d3); d:=a1*(b2*c3-b3*c2)-b1*(a2*c3-a3*c2)+c1*(a2*b3-a3*b2); dx:=-d1*(b2*c3-b3*c2)-b1*(-d2*c3+d3*c2)+c1*(-d2*b3+b2*d3); dy:=a1*(-c3*d2+c2*d3)+d1*(a2*c3-a3*c2)+c1*(-a2*d3+a3*d2); dz:=a1*(-b2*d3+b3*d2)-b1*(-a2*d3+a3*d2)-d1*(a2*b3-a3*b2); x:=dx/d; y:=dy/d; z:=dz/d; writeln('x=',x:8:12);readln; writeln('y=',y:8:12); writeln('z=',z:8:12);readln; End.



Documents

Blaise Pascal (1623–1662) - mathtoday.files.wordpress.com · កម វិធីគណនាប េផ្ស្គណិៗក កណវិវិទមយោត ោភាស. By