EGMO 2023 Problem 2

We are given an acute triangle $A B C$. Let $D$ be the point on its circumcircle such that $A D$ is a diameter. Suppose that points $K$ and $L$ lie on segments $A B$ and $A C$, respectively, and that $D K$ and $D L$ are tangent to circle $A K L$.

Show that line $K L$ passes through the orthocentre of $A B C$. The altitudes of a triangle meet at its orthocentre.

Figure 1: Diagram to solution 1