If x/(b+c-a)=y/(c+a-b)=z/(a+b-c);prove that x(by+cz-ax)=y(cz+ax-by)=z(ax+by-cz)  

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thilina-g | College Teacher | (Level 1) Educator

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If, `x/(b+c-a)=y/(c+a-b) =z/(a+b-c)`


`(b+c-a)/x=(c+a-b)y=(a+b-c)/z = k`


`b+c-a = kx`

`c+a-b = ky`

`a+b-c = kz`


This gives,

`c = k/2(x+y)`

`b = k/2(x+z)`

`a = k/2(y+z)`


Multiplying above three equations by z,y and x respectively.

`cz = k/2(x+y)z`

`by = k/2(x+z)y`

`ax = k/2(y+z)x`


Let `k/2 = 1/p` , then,


Rearranging gives,

`xz+yz = pcz`

`xy+yz = pby`

`xy+xz = pax`



`pby+pcz-pax = xy+yz+xz+yz-xy-xz`

`pby+pcz-pax = yz`

`by+cz-ax = (yz)/p`

Therefore, multiplying by x on both sides,

`x(by+cz-ax) = (xyz)/p`

Similarly, you can get.

`y(cz+ax-by) = (xyz)/p`

`z(ax+by-cz) = (xyz)/p`


Therefore, by combining above three,

`x(by+cz-ax) =y(cz+ax-by) =z(ax+by-cz) =(xyz)/p`



`x(by+cz-ax) =y(cz+ax-by) =z(ax+by-cz) `


pby+pcz-pax =

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