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?