2020 CIME I Problems/Problem 14

Problem 14

Let $ABC$ be a triangle with sides $AB = 5, BC = 7, CA = 8$. Denote by $O$ and $I$ the circumcenter and incenter of $\triangle ABC$, respectively. The incircle of $\triangle ABC$ touches $\overline{BC}$ at $D$, and line $OD$ intersects the circumcircle of $\triangle AID$ again at $K$. Then the length of $DK$ can be expressed in the form $\frac{m \sqrt n}{p}$, where $m, n, p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+p$.


Analytic geometry gives us \[DK=\frac{17\sqrt{57}}{19}.\] The answer is $93$.

See also

2020 CIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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All CIME Problems and Solutions

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