In the figure below, given a triangle
ABC and its orthic triangle DEF (AD, BE, and CF are the
altitudes of ABC). H is the orthocenter of triangle ABC, L, M, N
are the tangency points of the incircle of triangle DEF and DG and DK
are perpendicular to BE and CF respectively. Prove that D, G, N,
H, M, and K are concyclic points (lie on a circle).