Question 72
Given a segment $A B$ in the plane, choose on it a point $M$ different from $A$ and $B$. Two equilateral triangles $\triangle A M C$ and $\triangle B M D$ in the plane are constructed on the same side of segment $A B$. The circumcircles of the two triangles intersect in point $M$ and another point $N$. (The circumcircle of a triangle is the circle that passes through all three of its vertices.)
a. Prove that lines $A D$ and $B C$ pass through point $N$.
b. Prove that no matter where one chooses the point $M$ along segment $A B$, all lines $M N$ will pass through some fixed point $K$ in the plane.
a. Prove that lines $A D$ and $B C$ pass through point $N$.
b. Prove that no matter where one chooses the point $M$ along segment $A B$, all lines $M N$ will pass through some fixed point $K$ in the plane.