2013 AMC 12B Problems/Problem 24

Problem

Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$ , and $\overline{CN}$ is the angle bisector of $\angle{ACB}$ with $N$ on $\overline{AB}$ . Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$ . In addition $\triangle BXN$ is equilateral with $AC=2$ . What is $BN^2$ ?

$\textbf{(A)}\ \frac{10-6\sqrt{2}}{7} \qquad \textbf{(B)}\ \frac{2}{9} \qquad \textbf{(C)}\ \frac{5\sqrt{2}-3\sqrt{3}}{8} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{6} \qquad \textbf{(E)}\ \frac{3\sqrt{3}-4}{5}$

Solution

2013 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2013 AMC 12B Problems/Problem 24

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