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A4. Five Circles Theorem

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 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.

1 comment:

Kyle Smith said...

Hi Peter, I have been through your blog and have seen that you have wide background when it comes to mathematical computations, theories, observation and expertise. I would like to ask if you don't mind sharing your knowledge to our members?

We have this site named Semphi and it's pretty new and we have only few post. The purpose of the site is for education purposes specially for research and reference of the students. It would be a great help for us if you could share even a single post in relation to your expertise.

Hope you could grant us this request.

Semphi is located at

Thank you.


Kyle Smith