MathDB

Given a segment

AB

in the plane, choose on it a point

M

different from

A

and

B

. Two equilateral triangles

\triangle AMC

and

\triangle BMD

in the plane are constructed on the same side of segment

AB

. 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

AD

and

BC

pass through point

N

. (b) Prove that no matter where one chooses the point

M

along segment

AB

, all lines

MN

will pass through some fixed point

K

in the plane.

Problem Statement