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| == Problem ==
| | #REDIRECT [[2010_AMC_12A_Problems/Problem_8]] |
| Triangle <math>ABC</math> has <math>AB=2 \cdot AC</math>. Let <math>D</math> and <math>E</math> be on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, such that <math>\angle BAE = \angle ACD</math>. Let <math>F</math> be the intersection of segments <math>AE</math> and <math>CD</math>, and suppose that <math>\triangle CFE</math> is equilateral. What is <math>\angle ACB</math>?
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| <math>\textbf{(A)}\ 60^\circ \qquad \textbf{(B)}\ 75^\circ \qquad \textbf{(C)}\ 90^\circ \qquad \textbf{(D)}\ 105^\circ \qquad \textbf{(E)}\ 120^\circ</math>
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| == Solution ==
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| Let <math>\angle BAE = \angle ACD = x</math>.
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| <cmath>\begin{align*}\angle BCD &= \angle AEC = 60^\circ\\
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| \angle EAC + \angle FCA + \angle ECF + \angle AEC &= \angle EAC + x + 60^\circ + 60^\circ = 180^\circ\\
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| \angle EAC &= 60^\circ - x\\
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| \angle BAC &= \angle EAC + \angle BAE = 60^\circ - x + x = 60^\circ\end{align*}</cmath>
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| Since <math>\frac{AC}{AB} = \frac{1}{2}</math>, <math>\angle BCA = \boxed{90^\circ\ \textbf{(C)}}</math>
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| == See also ==
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| {{AMC10 box|year=2010|num-b=13|num-a=15|ab=A}}
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| [[Category:Introductory Geometry Problems]] | |
| {{MAA Notice}}
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