TITLE: Triangles whose vertices and center have all the same color
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: Actually, for any finite subset $P$ of the plane, there are $2^{\aleph_0}$ monochromatic scaled copies of $P$:
https://arxiv.org/abs/1304.3154
(see also the question Monochromatic point sets in two-colored plane)