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

(1)

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)

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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 following useful formulas.

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