smc

The question states that $$h+d=x+\sqrt{(x+h{)}^2+d^2}$$ We move $x$ to the left: $$h+d-x=\sqrt{(x+h{)}^2+d^2}$$ We square both sides: $$h^2+d^2+x^2-2xh-2xd+2hd=x^2+2xh+h^2+d^2$$ Cancelling and moving terms, we get: $$4xh+2xd=2hd$$ Factoring $x$: $$x(4h+2d)=2hd$$ Isolating for $x$: $$x=\frac{2hd}{4h+2d}=\frac{hd}{2h+d}$$ Therefore, the answer is $\boxed{\frac{hd}{2h+d}}$