Given a right-angled triangle ABC (∠C = 90). An arbitrary point M is selected on the leg BC.

Given a right-angled triangle ABC (∠C = 90). An arbitrary point M is selected on the leg BC. From point M, a perpendicular MN is drawn to the hypotenuse AB. Prove that ∠АNC = ∠AMC.

If you inscribe a right-angled triangle in a circle, then the hypotenuse of the triangle will lie on the diameter of the circle. Let us inscribe a right-angled triangle AFM (angle C = 90 °) in a circle, AM will be the diameter of the circle.

AM is the hypotenuse and triangle ANM. This means that the circle is also circumscribed about the triangle ANM.

Hence, points A, N, M and C lie on the same circle.

Angle ANC – inscribed, rests on the AC arc. AMC angle – inscribed, also rests on the AC arc.

Therefore, ∠АNC = ∠AMC (the inscribed angles based on the same arc are equal).