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Probability of Greater Area

Source:

February 19, 2008
probabilitygeometrysymmetryanalytic geometry

Problem Statement

A point P P is chosen at random in the interior of equilateral triangle ABC ABC. What is the probability that △ABP \triangle ABP has a greater area than each of △ACP \triangle ACP and △BCP \triangle BCP? <spanclass=′latex−bold′>(A)</span> 16<spanclass=′latex−bold′>(B)</span> 14<spanclass=′latex−bold′>(C)</span> 13<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 23 <span class='latex-bold'>(A)</span>\ \frac{1}{6} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{4} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{3} \qquad <span class='latex-bold'>(D)</span>\ \frac{1}{2} \qquad <span class='latex-bold'>(E)</span>\ \frac{2}{3}
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