Japanese Circumscribed Quadrilateral Theorem

Dedication: This little page is a small tribute to John Rigby from Cardiff University who passed away at the end of 2014. I had the pleasure of attending several of his brilliant, insightful and inspiring talks on geometry at annual meetings of the Mathematical Association, and also personally communicating with him on more than one occasion. This link has a short obituary by Gerry Leversha that was published in the Mathematical Gazette (2015).


The following remarkably beautiful Japanese Temple Geometry theorem is given and proved in Fukagawa, H. & Rigby, J. (2002). Traditional Japanese Mathematics Problems of the 18th and 19th Centuries. Singapore: SCT Publishing, p. 22.

"The products of the radii of the excircles on each pair of opposite sides of a circumscribed quadrilateral are equal."

Explore the result by dragging any of the red tangent points I, J, K or L to change the shape of the circumscribed quadrilateral.


Japanese Circumscribed Quadrilateral Theorem

Though not mentioned in the above publication, the result beautifully generalizes to 2n-gons circumscribed around a circle in relation to the products of the radii of the excircles on alternate sides. Click on the links provided in the sketch above to navigate to the hexagon and octagon cases.

Can you explain why (prove that) the main result and its generalization(s) are true? If so, can you explain (prove) the result in different ways?

Solution: The following handwritten solution, using trigonometry, was received from Johan Meyer, Dept. of Math, University of Free State Solution 1.

More Solutions, Variations and Generalizations:
1) Read the 1989 paper by Hiroshi Okumura, "A Theorem on Tangent Cycles".
2) Read the 2018 paper by John Silvester, "Variations on a Japanese Temple theorem".

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Created by Michael de Villiers, 20 June 2016; modified 1 Oct 2016; 30 Jan 2017, 21 October 2018.