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.
Clawson, J. W. "A Chain of Circles Associated with the 5-Line." Amer. Math. Monthly61, 161-166, 1954.Li, K. Y. "Concyclic Problems." Math. Excalibur6-1, 1-2, 2001. http://www.math.ust.hk/excalibur/v6_n1.pdf.Miquel, A. "Mémoire de Géométrie." J. de mathématiques pures et appliquées de Liouville1, 485-487, 1838.
