MathDB

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}

, when

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

Problem Statement