Van Aubel's Theorem (GEOM-10)

The Futility Closet reminded me today of one of the greatest 'how is that possible' theorems in plane geometry: Van Aubel's Theorem.[br]http://www.futilitycloset.com/2015/08/07/van-aubels-theorem/[br][br]What is going on here? Why is it true?
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Information: Van Aubel's Theorem (GEOM-10)