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# Japanese Theorem

Let a convex cyclic polygon be triangulated in any manner, and draw the incircle to each triangle so constructed. Then the sum of the inradii is a constant independent of the triangulation chosen. This theorem can be proved using Carnot's theorem. In the above figures, for example, the inradii of the left triangulation are 0.142479, 0.156972, 0.232307, 0.498525, and the inradii of the right triangulation are 0.157243, 0.206644, 0.312037, 0.354359, giving a sum of 1.03028 in each case.

According to an ancient custom of Japanese mathematicians, this theorem was a Sangaku problem inscribed on tablets hung in a Japanese temple to honor the gods and the author in 1800 (Johnson 1929).

The converse is also true: if the sum of inradii does not depend on the triangulation of a polygon, then the polygon is cyclic.

Carnot's Theorem, Cyclic Polygon, Incircle, Inradius, Sangaku Problem, Triangulation

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## References

Hayashi, T. "Sur un soi-disant théorème chinois." Mathesis 6, 257-260, 1906.Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 24-26, 1985.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 193, 1929.Lambert, T. "The Delaunay Triangulation Maximizes the Mean Inradius." Proc. Sixth Canadian Conf. Comput. Geometry. Saskatoon, Saskatchewan, Canada, pp. 201-206, Aug. 1994.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 125, 1991.

Japanese Theorem

## Cite this as:

Weisstein, Eric W. "Japanese Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JapaneseTheorem.html