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.