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# Miquel's Pentagram Theorem

Consider a convex pentagon and extend the sides to a pentagram. Externally to the pentagon, there are five triangles. Construct the five circumcircles. Each pair of adjacent circles intersect at a vertex of the pentagon and a second point. Then Miquel's pentagram theorem states that these five second points are concyclic.

This theorem is sometimes referred to as Jiang Zemin's problem, as this former president of China talked about the theorem in the end of 1999 as he visited Macau.

Miquel Five Circles Theorem

This entry contributed by Floor van Lamoen

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## References

Clawson, J. W. "A Chain of Circles Associated with the 5-Line." Amer. Math. Monthly 61, 161-166, 1954.Li, K. Y. "Concyclic Problems." Math. Excalibur 6-1, 1-2, 2001. http://www.math.ust.hk/excalibur/v6_n1.pdf.Miquel, A. "Mémoire de Géométrie." J. de mathématiques pures et appliquées de Liouville 1, 485-487, 1838.

## Referenced on Wolfram|Alpha

Miquel's Pentagram Theorem

## Cite this as:

van Lamoen, Floor. "Miquel's Pentagram Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/MiquelsPentagramTheorem.html