Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Find the answer

Let $A B$ be a segment of length 2 with midpoint $M$ . Consider the circle with center $O$ and radius $r$ that is externally tangent to the circles with diameters $A M$ and $B M$ and internally tangent to the circle with diameter $A B$ . Determine the value of $r$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Let $X$ be the midpoint of segment $A M$ . Note that $O M \perp M X$ and that $M X=\frac{1}{2}$ and $O X=\frac{1}{2}+r$ and $O M=1-r$ . Therefore by the Pythagorean theorem, we have $O M^{2}+M X^{2}=O X^{2} \Longrightarrow(1-r)^{2}+\frac{1}{2^{2}}=\left(\frac{1}{2}+r\right)^{2}$ which we can easily solve to find that $r=\frac{1}{3}$ .

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