How to use triangle similarity to show that $a^2=y \cdot c$?

Question

Is the image below correct regarding the angles and that $\angle A= \angle B$ ? But there is 3 triangles, how would you show that $a^2=y\cdot c$ ? Not sure if i was thinking correct (all other than blue color), but i don't see the connections when its all put together, very easy when we have 2 triangles side by side where we get the difference between similar side lengths but it doesn't matter which side we divide with as long as it's one side of another triangle, or?

Answer 1

In your diagram $\triangle ABC$, $\triangle CBD$ and $\triangle ACD$ are similar right-angled triangles so $\frac{c}{a}=\frac{a}{y}$ and so $a^2=cy$

This does not suggest $\angle A= \angle B$, though it does suggest $\angle A+\angle B=\frac{\pi}{2}$, or $=90^\circ$ if you prefer

Answer 2

Hint: $\;\dfrac{y}{a}=\cos B=\dfrac{a}{c}\,$.

You could eliminate the $\,\cos B\,$ step by rewriting it as similarity between $\triangle DBC$ and $\triangle CBA$.