Mock Geometry AIME 2011 Problems/Problem 6

Problem

Three points $A,B,C$ are chosen at random on a circle. The probability that there exists a point $P$ inside an equilateral triangle $A_1B_1C_1$ such that $PA_1=BC,PB_1=AC,PC_1=AB$ can be expressed in the form $\frac{m} {n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

The problem asks for the probability that point $P$ is inside an equilateral triangle $A_1B_1C_1$ . Let $x$ , $y$ , and $z$ be the three distances from point $P$ to each of the vertices, with $x$ being the longest distance. Let's consider the case in which point $P$ is actually on the line: $[asy] unitsize(1cm); draw((0,3sqrt(3))--(3,0)--(12,0)--cycle); draw((0, 18*sqrt(3))--(18, 0)); draw((18, 0)--(36, 18*sqrt(3))); [/asy]$

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Mock Geometry AIME 2011 Problems/Problem 6

Problem

Solution