Triangle height question dividing edge

Question

In a right triangle ABC, with right angle at C, H is the foot of the height from C. Show that $\overline{CH}^2 =\overline{AH} · \overline{BH}$.

I really have no idea to do this.

Answer 1

Note that the $\angle BCH=\angle CAB$ and that $\angle CHB=\angle CHA$ as $\triangle ABC$ is a right triangle. So we have that $$\triangle CHA \sim \triangle BHC$$ Since two angles are the same. So $$\frac{\overline{CH}}{\overline{AH}}=\frac{\overline{BH}}{\overline{CH}}$$ Thus we have $$\overline {CH}^2=\overline {AH} \times \overline {BH}$$