Let ABC be an acute triangle. Tangents on the circumscribed circle of triangle ABC at points B and C intersect at point T. Let D and E be a foot of the altitudes from T onto AB and AC and let M be the midpoint of BC. Prove:
A) Prove that M is the orthocenter of the triangle ADE.
B) Prove that TM cuts DE in half. JBMO TSTgeometrycircumcirclenational olympiad