3500.21 – Midpoint of a Chord?

The radius of the large circle is twice the radius of the small one. The circles are internally tangent at A. B is any point on the large circle. AB intersects the small circle at M. Is M the midpoint of AB?

Solution

Draw diameter AD, which passes through C, the center of the large circle. Draw MC and BD. $\triangle$ AMC and $\triangle$ ABD are right triangles that are similar (AA). Thus, since AD = 2AC, we have AB = 2AM, so M is indeed the midpoint of AB.