Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$. Tangent lines to $(O)$ at $B,C$ meet at $T$. A line passes through $T$ cuts segments $AB$ at $D$ and cuts ray $CA$ at $E$. Take $M$ as midpoint of $DE$ and suppose that $MA$ cuts $(O)$ again at $K$. Prove that $(MKT)$ is tangent to $(O)$.
Problem
Source: 2021 Saudi Arabia Training Lists p17 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests
Tags: geometry, tangent circles, circumcircle