Question 39
Let $AB\Gamma$ an equilateral triangle of side $\alpha$. Let $\Delta$, $E$ and $Z$ the midpoints of the sides $AB$, $B\Gamma$ and $\Gamma A$, respectively. Let $H$ the symmetric point of $\Delta$ with respect to the line $B\Gamma$. We color the points $A$, $B$, $\Gamma$, $\Delta$, $E$, $Z$, $H$ using one of the two colors $\kappa$ = red and $\mu$ = blue.
a. Find how many equilateral triangles are defined with vertices from the seven points $A$, $B$, $\Gamma$, $\Delta$, $E$, $Z$, $H$.
b. Prove that, if the points $B$ and $E$ will be colored with the same color, then for every coloring of the remaining points there exists an equilateral triangle with vertices from the points $A$, $B$, $\Gamma$, $\Delta$, $E$, $Z$, $H$ whose all vertices have the same color.
c. Can we have the same conclusion, if the points $B$ and $E$ will be colored by different colors?
a. Find how many equilateral triangles are defined with vertices from the seven points $A$, $B$, $\Gamma$, $\Delta$, $E$, $Z$, $H$.
b. Prove that, if the points $B$ and $E$ will be colored with the same color, then for every coloring of the remaining points there exists an equilateral triangle with vertices from the points $A$, $B$, $\Gamma$, $\Delta$, $E$, $Z$, $H$ whose all vertices have the same color.
c. Can we have the same conclusion, if the points $B$ and $E$ will be colored by different colors?