Question

Solve the following equations :
$x^2=a+(y-z{)}^2,y^2=b+(z-x{)}^2,z^2=c+(x-y{)}^2.$

Answer 1

The equations as can be rewritten as
$x^2-{(y-z)}^2=a$ or $(x+y-z)(x+z-y)=a$
Similarly $(y+z-x)(y+x-z)=b$ and $(z+x-y)(z+y-x)=c.$
Multiplying and taking square root, we get
$(x+y-z)(z+x-y)(y+z-x)=\pm \sqrt{abc}$
Hence $y+z-x=\pm \sqrt{\left(\frac{bc}{a}\right)},$
$z+x-y=\pm \sqrt{\left(\frac{ca}{b}\right)},$
and $x+y-z=\pm \sqrt{\left(\frac{ab}{c}\right)}.$
Adding last two equations, we get
$x=\pm \frac{1}{2}\{\sqrt{\frac{ca}{b}}+\sqrt{\frac{ab}{c}}\}.$
Similarly, $y=\pm \frac{1}{2}\{\sqrt{\frac{bc}{a}}+\sqrt{\frac{ab}{c}}\}$
and $z=\pm \frac{1}{2}\{\sqrt{\frac{bc}{a}}+\sqrt{\frac{ca}{b}}\}$.