Intuition why Cauchy-Schwartz inequality implies triangle inequality?

Question

This question is not about an intuition of why the Cauchy-Schwartz inequality is true, nor why the triangle inequality is true.

Nor am I asking for a formal proof of the triangle inequality from the Cauchy-Schwartz inequality.

Rather, I'd like to have a geometric intuition for why the one follows from the other. (I can prove it algebraically).

Answer 1

It's basically what algebra says:

1. The square constructed on the opposite side $c$ has the same area as the sum of the square on the other two sides ("plus or", if we do not use signed lengths) minus twice the rectangle with sides $a$ and the projection of $b$ on $a$.

2. The square constructed on the sum of the sides $a+b$ can be divided into a square of side $a$, a square of side $b$ and two rectangles of sides $a,b$.

3. Cauchy-Schwarz says which of the rectangles are the largest.