triangle.md

The Right Triangle

Base Geometry

Let the right triangle hypothenuse be aligned with the coordinate system x-axis. The vector loop closure equation running counter-clockwise then reads

$$a{\bold e}\alpha + b\tilde{\bold e}\alpha + c{\bold e}_x = \bold 0$$

with

$${\bold e}\alpha = \begin{pmatrix}\cos\alpha\ \sin\alpha\end{pmatrix} \quad and \quad {\tilde\bold e}\alpha = \begin{pmatrix}-\sin\alpha\ \cos\alpha\end{pmatrix}$$

Resolving for the hypothenuse part $c{\bold e}_x$ in the loop closure equation (1)

$$-c{\bold e}x = a{\bold e}\alpha + b\tilde{\bold e}_\alpha$$

and squaring

finally results in the Pythagorean theorem (2)

$$c^2 = a^2 + b^2$$ (2)

More Triangle Stuff

Introducing the hypothenuse segments $p={\bold a}\cdot{\bold e}_x$ and $q={\bold b}\cdot{\bold e}_x$, we can further obtain the following useful formulas.

segment p	segment q	height h	area
$cp = a^2$	$cq = b^2$	$pq = h^2$	$ab = ch$