An acute triangle ABC is given and let H be its orthocenter. Let ω be the circle through B, C and H, and let Γ be the circle with diameter AH. Let X=H be the other intersection point of ω and Γ, and let γ be the reflection of Γ over AX. Suppose γ and ω intersect again at Y=X, and line AH and ω intersect again at Z=H. Show that the circle through A,Y,Z passes through the midpoint of segment BC. geometrygeometry proposed