Question 14
Let $ABCD$ be a tetrahedron such that $AB = CD = \sqrt{41},$ $AC = BD = \sqrt{80},$ and $BC = AD = \sqrt{89}.$ There exists a point $I$ inside the tetrahedron such that the distances from $I$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{m\sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m + n + p.$