$ABCD$ is a trapezoid with $BC\parallel AD$ and $BC = 2AD$. Point $M$ is chosen on the side $CD$ such that $AB = AM$. Prove that $BM \perp CD$. Proposed by Bogdan Rublov
Source: Kyiv City MO 2024 Round 1, Problem 11.2
Tags: geometry, trapezoid
$ABCD$ is a trapezoid with $BC\parallel AD$ and $BC = 2AD$. Point $M$ is chosen on the side $CD$ such that $AB = AM$. Prove that $BM \perp CD$. Proposed by Bogdan Rublov