AB and CM are common tangents to two circles.
M is the midpoint of AB.
Angle ACB is a right angle.
AM = CM and BM = CM so AM = BM. So M is the midpoint of AB.
Triangle AMC is isosceles, so angles MAC and MCA are equal, call them alpha.
Triangle BMC is isosceles, so angles MBC and MCB are equal, call them beta.
Sum the angles in the triangle ABC, 2.alpha + 2.beta = 180, so alpha + beta = 90 = angle ACB.