Side Divider Theorem for a Circumscribed/Tangential 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 angle dividers of a cyclic quadrilateral go to Angle Divider Theorem for a Cyclic Quadrilateral
Proofs of this result, its dual & of the generalizations, are given in Some Adventures in Euclidean Geometry as well as generalizations to circumscribed 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.
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Updated by Michael de Villiers, 24 March 2012; updated 3 September 2020 with WebSketchpad.