Let ABC be an equilateral triangle of side length a, and consider D, E and F the midpoints of the sides (AB),(BC), and (CA), respectively. Let H be the the symmetrical of D with respect to the line BC. Color the points A,B,C,D,E,F,H with one of the two colors, red and blue.[*] How many equilateral triangles with all the vertices in the set {A,B,C,D,E,F,H} are there?
[*] Prove that if points B and E are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set {A,B,C,D,E,F,H} and having the same color.
[*] Does the conclusion of the second part remain valid if B is blue and E is red?
Greecegeometrycombinatorics