Question

$Daca x, y si z sunt lungimile laturilor unui triunghi si y+z-x/x=z+x-y/y=x+y-z/z, calculati (x+y)(y+z)(z+x)/xyz.$

Answer 1

Răspuns:

(x+y)(y+z)(x+z)/(xyz)=8

Explicație pas cu pas:

x, y, z lungimile laturilor unui triunghi

(y+z-x)/x=(x+z-y)/y=(x+y-z)/z

(y+z)/x-x/x=(x+z)/y-y/y=(x+y)/z-z/z

(y+z)/x -1=(x+z)/y -1=(x+y)/z -1

(y+z)/x=(x+z)/y=(x+y)/z

=> (y+z)/x=(x+z)/y=(x+y)/z=(y+z+x+z+x+y)/(x+y+z)

(y+z)/x=(x+z)/y=(x+y)/z=2(x+y+z)/(x+y+z)

(y+z)/x=(x+z)/y=(x+y)/z=2

y+z=2x

x+z=2y

x+y=2z

(x+y)(y+z)(x+z)/(xyz)=2x*2y*2z/(xyz)=2*2*2=8