2012 AIME II Problems/Problem 15

Problem 15

Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$ , $BC=7$ , and $AC=3$ . The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$ . Let $\gamma$ be the circle with diameter $\overline{DE}$ . Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$ . Then $AF^2 = \frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Use angle bisector theorem to find $CD=21/8$ , $BD=35/8$ , and $AD=15/8$ . Use Power of the Point to find $ED=49/8$ , and so $AE=8$ . Use law of cosines to find \angle CAD = \pi/3 }

2012 AIME II (Problems • Answer Key • Resources)
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2012 AIME II Problems/Problem 15

Problem 15

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