Geometric justification of the inner product

Question

In the picture $a$, $b$,$c$ and $d$ are the lengths of the segments.

Why $ab$ = $cd$ ? Thanks

ps: it's the geometric justification of the inner product

Answer 1

Your claim only holds if $b$ and $d$ are the whole triangle side lengths of the larger of the two similar right triangles draw on the angle at the left, rather than - as they appear to be - the incremental lengths. Because then, due to the triangle similarity, $\dfrac{a}{c}=\dfrac{d}{b}$ and your result would follow.

Answer 2

$$\frac{a}{c+d}=\frac{c}{a+b}$$ and we can not get which you wish.