DE-MO 2022 Problem 5

Let $ABC$ be an equilateral triangle with circumcircle $k$ . A circle $q$ touches $k$ from outside in a point $D$ , where the point $D$ on $k$ is chosen so that $D$ and $C$ lie on different sides of the line $AB$ . We now draw tangent lines from $A,B$ and $C$ to the circle $q$ and denote the lengths of the respective tangent line segments by $a,b$ and $c$ .

Prove that $a+b=c$ .