Let the right triangle hypothenuse be aligned with the coordinate system x-axis.
-The vector loop closure equation running counter-clockwise then reads
-aeα+be~α+cex=0(1)
with
-eα=(cosαsinα)ande~α=(−sinαcosα)
Resolving for the hypothenuse part cex in the loop closure equation (1)
-−cex=aeα+be~α
and squaring
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finally results in the Pythagorean theorem (2)
-c2=a2+b2(2)
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More Triangle Stuff
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Introducing the hypothenuse segments p=a⋅ex and q=b⋅ex, we can further obtain the following useful formulas.