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Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 11"

 
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<math>11.</math> <math>ABC</math> is an acute triangle with perimeter <math>60</math>. <math>D</math> is a point on <math>\overline{BC}</math>. The circumcircles of triangles <math>ABD</math> and <math>ADC</math> intersect <math>\overline{AC}</math> and <math>\overline{AB}</math> at <math>E</math> and <math>F</math> respectively such that <math>DE = 8</math> and <math>DF = 7</math>. If <math>\angle{EBC} \cong \angle{BCF}</math>, then the value of <math>\frac{AE}{AF}</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>.
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==Problem==
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<math>ABC</math> is an acute triangle with perimeter <math>60</math>. <math>D</math> is a point on <math>\overline{BC}</math>. The circumcircles of triangles <math>ABD</math> and <math>ADC</math> intersect <math>\overline{AC}</math> and <math>\overline{AB}</math> at <math>E</math> and <math>F</math> respectively such that <math>DE = 8</math> and <math>DF = 7</math>. If <math>\angle{EBC} \cong \angle{BCF}</math>, then the value of <math>\frac{AE}{AF}</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>.
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==Solution==
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{{solution}}
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==See also==

Revision as of 07:31, 14 February 2008

Problem

$ABC$ is an acute triangle with perimeter $60$. $D$ is a point on $\overline{BC}$. The circumcircles of triangles $ABD$ and $ADC$ intersect $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$ respectively such that $DE = 8$ and $DF = 7$. If $\angle{EBC} \cong \angle{BCF}$, then the value of $\frac{AE}{AF}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Solution

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See also

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