BALKAN-SL 2020 G2

Let $G, H$ be the centroid and orthocentre of $\vartriangle ABC$ which has an obtuse angle at $\angle B$ . Let $\omega$ be the circle with diameter $AG$ . $\omega$ intersects  $\odot(ABC)$ again at $L \ne A$ . The tangent to $\omega$ at $L$ intersects  $\odot(ABC)$ at $K \ne L$ . Given that $AG = GH$ , prove $\angle HKG = 90^o$ . Sam Bealing, United Kingdom