A Fun Proof of Van Aubel's Theorem.

Van Aubel's theorem isn't much more than a curious geometrical construction, but the more you think about it, the more interesting it seems. There are a few published proofs out there on the internet. Most involve constructing similar triangles, but the one that fascinated me involved placing the quadrilateral in the complex plane and then doing complex algebra to show that multiplying one of the lines by i (i.e. rotating it by 90 degrees) resulted in an equation that equals zero, proving equivalence of the two lines. I have essentially used the same method, substituting abstract vectors for complex numbers, which makes it more conducive to animation.

Apologies for my terrible guess at how to pronounce of "Henri Van Aubel". I'm not French, and neither was he!

Source of the complex number proof: https://www.i-repository.net/il/user_contents/02/G0000031Repository/repository/keidaironshu_063_006_131-138.pdf

Animation was done with Desmos Graphing Calculator. You can interact with it here: https://www.desmos.com/calculator/lldmzybkbc

Music: “Fifth Avenue Stroll”, iMovie Song (https://youtu.be/UI6YgHidevk)

Corrections:

0:59 It is more accurate to say that Van Aubel was Belgian, not Dutch. He lived and taught in Belgium, and wrote his theorems in French. I'm not sure what my source was for his being Dutch; I cannot find it now. It seems likely that he was of Dutch ancestry at least.

Видео A Fun Proof of Van Aubel's Theorem. канала MathyJaphy