A plane is colored with two colors. It's an easy exercise to prove that it's alaways possible to find an equilateral triangle whose vertices have all the same color.

Does anyone know any proof of reference for the following problem?

A plane is colored with two colors. Is it always possible to find a triangle whose vertices and his center C have all the same color? Consider four different cases:

i) C is the incenter
ii) C is the circumcenter
iii) C is the orthocenter
iv) C is the barycenter.

Any help would be appreciated.

A plane is colored with two colors. It's an easy exercise to prove that it's always possible to find an equilateral triangle whose vertices have all the same color.

Does anyone know any proof or reference for the following problem?

A plane is colored with two colors. Is it always possible to find a triangle whose vertices and center C have all the same color? Consider four different cases:

i) C is the incenter
ii) C is the circumcenter
iii) C is the orthocenter
iv) C is the barycenter.

Any help would be appreciated.