Thébault's theorem

Then construct two additional circles, each tangent to AM, BC, and to the circumcircle.

[3][4] Until 2003, academia thought this third problem of Thébault the most difficult to prove.

It was published in the American Mathematical Monthly in 1938, and proved by Dutch mathematician H. Streefkerk in 1973.

However, in 2003, Jean-Louis Ayme discovered that Sawayama Yuzaburo [ja], an instructor at The Central Military School of Tokyo, independently proposed and solved this problem in 1905.

[5] An "external" version of this theorem, where the incircle is replaced by an excircle and the two additional circles are external to the circumcircle, is found in Shay Gueron (2002).