Prove the triangle is equilateral

Question

HINTS ONLY please.

This is very confusing right off the bat. My guess was that we show the angle $C, M, N$ are all $60^{\text{o}}.$

But I am having difficulty doing as as none of the givens have led me to any success.

Answer 1

Hint. BCE and ACD are congruent.

Answer 2

This is straightforward (if a little messy) with coordinates.

Suppose without loss of generality that $A,B,C$ have Cartesian coordinates $(-1/2, \sqrt{3}/2)$; $(-1,0)$; and $(0,0)$ respectively. Let the side length of $ECD$ be $x$. Then you can find the coordinates for $D$ and $E$, and then for $M$ and $N$, and then use the distance formula to see that the three sides of the triangle under consideration are all equal (they all turn out to be $\sqrt{1+x+x^2}$).

Answer 3

Hint: find a suitable rotation around $C$.