Angle Divider Theorem for a Cyclic Quadrilateral
For a proof of the quadrilateral case, read my AMESA 1996 paper An interesting Side-Angle Duality in Geometry.
For a side-angle dual of this result involving the side dividers of a circumscribed/tangential quadrilateral go to Side Divider Theorem for a Circumscribed Quadrilateral
Proofs of this result, its dual & of the generalizations, are given in Some Adventures in Euclidean Geometry as well as generalizations to cyclic polygons with an even or odd number of sides. The book is available for purchase as a downloadable PDF, printed book or from iTunes for your iPhone, iPad, or iPod touch, and on your computer with iTunes.
Updated by Michael de Villiers, 24 March 2012; updated 3 September 2020 with WebSketchpad.