Point $M$ on side $AB$ of quadrilateral $ABCD$ is such that quadrilaterals $AMCD$ and $BMDC$ are circumscribed around circles centered at $O_1$ and $O_2$ respectively. Line $O_1O_2$ cuts an isosceles triangle with vertex $M$ from angle $CMD$. prove that $ABCD$ is a cyclc quadrilateral.
Problem
Source: 2022 Saudi Arabia January Camp Test 1.2 BMO + EGMO TST
Tags: geometry, tangential quadrilateral, cyclic quadrilateral, Concyclic