MathNet 2015-part-03 — 191 problems (EN–VI) Question 129

Let $ABCD$ be a convex quadrilateral such that $AB = AC = AD$ and $BC \lt CD$. The bisector of the angle $\widehat{BAD}$ meets $CD$ internally at $M$ and the extension of $BC$ at $N$. Prove that

a. the quadrilateral $ABCM$ is cyclic;

b. the triangles $ANB$ and $ABM$ are similar.