MathDB

The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths

2\sqrt3

5

, and

\sqrt{37}

, as shown, is

\tfrac{m\sqrt{p}}{n}

, where

m

n

, and

p

are positive integers,

m

and

n

are relatively prime, and

p

is not divisible by the square of any prime. Find

m+n+p

. [asy] size(5cm); pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); real t = .385, s = 3.5*t-1; pair R = A*t+B*(1-t), P=B*s; pair Q = dir(-60) * (R-P) + P; fill(P--Q--R--cycle,gray); draw(A--B--C--A^^P--Q--R--P); dot(A--B--C--P--Q--R); [/asy]

Problem Statement