MathDB

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?

Problem Statement