From The British Mathematical Olympiad, Paper 2, 1998, Question 3:
Suppose $x, y, z$ are positive integers satisfying the equation $\frac{1}x - \frac{1}y = \frac1{z}$ and let $h$ be the highest common factor of $x, y, z$.
Prove that $hxyz$ is a perfect square.
Prove also that $h(y−x)$ is a perfect square.
Here is my attempt for part 1:
$\frac{1}x - \frac{1}y = \frac1{z}$
$yz-xz = xy\\z(y-x) = xy$
$x = c_1h\\y= c_2h\\z = c_3h$
$c_3h(c_2h-c_1h) = xy\\c_3h^2(c_2-c_1) = xy\\h^2(c_3(c_2-c_1)) = xy\\zh^3(c_3(c_2-c_1)) = hxyz\\h^4(c_3^2(c_2-c_1)) = hxyz$
And for part 2:
$h(y−x)=h^2(c_2-c_1)$
I need some hints on how to prove $c_2-c_1$ is a perfect square which I believe is the only step left; if any of my logic or presentation is wrong, please tell me as well.