Please help with this problem

Expanding the terms, 

$(x^2+2xh+h^2)+(y^2+2yh+y^2)=(a^2+2ab+b^2)$

Using Pythagorean Theorem,

$x^2+h^2=a^2\\ y^2+h^2=b^2$

We could subsitute the values in, and rewrite the equation

$a^2+2xh+b^2+2yh=a^2+2ab+b^2\\ 2xh+2yh=2ab\\ h(x+y)=ab$

$[ABC]=\frac12 AB\cdot CH = \frac12 BC \cdot AC\\ \frac12 (x+y)h=\frac12 ab\\ h(x+y)=ab$

I hope this helped,

Gavin