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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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Remark that since <math>ABDE</math> is cyclic we have <math>\angle CED=\angle DBF</math>, and similarly <math>\angle BFD=\angle DCE</math>. Therefore by AA similarity <math>\triangle DBF\sim\triangle DEC</math>. Thus there exists a spiral similarity sending <math>B</math> to <math>E</math> and <math>F</math> to <math>D</math>, so by a fundamental theorem of spiral similarity <math>\triangle BDE\sim\triangle FDC</math>. The angle equality condition gives <math>\angle CFD=\angle EBD=\angle DCF</math>, so <math>\triangle CDF</math> is isosceles and <math>DC=7</math>. Similarly, <math>BD=8</math>. Finally, note that the congruent side lengths actually imply <math>\triangle DBF=\triangle DEC</math>, so <math>EC=BF</math>.
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Let <math>x=CE=BF</math> and <math>b=AC</math>. Remark that from the perimeter condition <math>AB=45-b</math>. Now from Power of a Point we have the system of two equations <cmath>\begin{cases}7\cdot 15&=xb,\\8\cdot 15&=x(45-b).\end{cases}</cmath> Expanding the second equation and rearranging variables gives <math>45x=8\cdot 15+bx=15^2\implies x=5</math>. Back-substitution yields <math>AC=21</math> and consequently <math>AB=24</math>. Thus <math>AE=16</math> and <math>AF=19</math>, so the desired ratio is <math>\tfrac{16}{19}\implies\boxed{035}</math>.
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==See Also==
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{{Mock AIME box|year=Pre 2005|n=3|num-b=10|num-a=12}}

Latest revision as of 14:32, 22 March 2015

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

Remark that since $ABDE$ is cyclic we have $\angle CED=\angle DBF$, and similarly $\angle BFD=\angle DCE$. Therefore by AA similarity $\triangle DBF\sim\triangle DEC$. Thus there exists a spiral similarity sending $B$ to $E$ and $F$ to $D$, so by a fundamental theorem of spiral similarity $\triangle BDE\sim\triangle FDC$. The angle equality condition gives $\angle CFD=\angle EBD=\angle DCF$, so $\triangle CDF$ is isosceles and $DC=7$. Similarly, $BD=8$. Finally, note that the congruent side lengths actually imply $\triangle DBF=\triangle DEC$, so $EC=BF$.

Let $x=CE=BF$ and $b=AC$. Remark that from the perimeter condition $AB=45-b$. Now from Power of a Point we have the system of two equations \[\begin{cases}7\cdot 15&=xb,\\8\cdot 15&=x(45-b).\end{cases}\] Expanding the second equation and rearranging variables gives $45x=8\cdot 15+bx=15^2\implies x=5$. Back-substitution yields $AC=21$ and consequently $AB=24$. Thus $AE=16$ and $AF=19$, so the desired ratio is $\tfrac{16}{19}\implies\boxed{035}$.

See Also

Mock AIME 3 Pre 2005 (Problems, Source)
Preceded by
Problem 10 		Followed by
Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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