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AB and CM are common tangents to two circles.

Show that:

M is the midpoint of AB. 

Angle ACB is a right angle. Unable to Display Picture.

 Dec 16, 2020
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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.

 Dec 18, 2020

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