Given two vectors $u = (x_1,y_1)$ and $v=(x_2,y_2)$ there is the dot product $u\cdot v = x_1 x_2 + y_1 y_2$. This has the geometric interpretation $u \cdot v = |u||v|\cos\theta$. The standard proof of that geometric interpretation goes through the law of cosines which itself is proved with the pythagorean theorem or the techniques used in the proof of the pythagorean theorem. For example, the following image illustrates a proof of the law of cosines using the orthocenter.
But how can one prove $x_1 x_2 + y_1 y_2 = |u||v|\cos\theta$ directly geometrically? There should be some way to prove it using my horribly drawn diagram below. The top left shaded area is $|u||v|\cos\theta$. And the other two shaded areas are $x_1 x_2$ and $y_1 y_2$. How does one prove that using geometry and trigonometry like in the first diagram for the law of cosines.