A4. Five Circles Theorem

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Harold Jacobs presents the fascinating Five Circles Theorem on page 568 of his excellent high school text: Geometry: Seeing, Doing, Understanding (3rd ed.). It states if one starts with a cyclic quadrilateral ABCD, draws the diagonals AC and BD, inscribes a circle in each of the 4 triangles produced, and connects the centers of these circles, then the quadrilateral EFGH produced is a rectangle!

Peter Renz, an editor of Jacobs, called this theorem to my attention and mentioned that the proof that Jacobs gives in the Teacher's Guide uses transformational geometry. He asked if I could find a more elementary proof.

I struggled a bit with this, but finally came up a proof which I have posted at http://www.scribd.com/doc/98720253. I found the Geometer's Sketchpad computer program to be invaluable in helping me discovering geometric truths which I was able to prove and put together to create the proof.

If you are good at geometry, you may want to see if you can come up with a proof on your own.