CAPS 2019 Problem 6

Let $ABC$ be an acute triangle with $AB<AC$ and $\angle BAC=60^{\circ}$ . Denote its altitudes by $AD,BE,CF$ and its orthocenter by $H$ . Let $K,L,M$ be the midpoints of sides $BC,CA,AB$ , respectively. Prove that the midpoints of segments $AH, DK, EL, FM$ lie on a single circle.