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Difference between revisions of "Mock AIME 1 2007-2008 Problems/Problem 11"

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== Solution ==
 
== Solution ==
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<!-- <center><asy></asy></center> -->
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<center><asy>
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pointpen = black; pathpen = black;
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pair A=(0,0),B=(23,0),C=IP(Circle(A,24),Circle(B,25));
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D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle);
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pair D=(B+C)/2,E=(A+C)/2,F=(A+B)/2;
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D(circumcircle(MP("D",D),MP("E",E),C));
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D(circumcircle(B,MP("F",F),D));
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D(circumcircle(A,F,E));
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</asy></center>
 
{{solution}}
 
{{solution}}
  

Revision as of 17:39, 24 April 2008

Problem

$\triangle DEF$ is inscribed inside $\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\triangle DEC, \triangle BFD, \triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB = 23, BC = 25, AC=24$, and $\stackrel{\frown}{BF} = \stackrel{\frown}{EC},\ \stackrel{\frown}{AF} = \stackrel{\frown}{CD},\ \stackrel{\frown}{AE} = \stackrel{\frown}{BD}$. The length of $BD$ can be written in the form $\frac mn$, where $m$ and $n$ are relatively prime integers. Find $m+n$.

Solution

[asy] pointpen = black; pathpen = black; pair A=(0,0),B=(23,0),C=IP(Circle(A,24),Circle(B,25)); D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle); pair D=(B+C)/2,E=(A+C)/2,F=(A+B)/2; D(circumcircle(MP("D",D),MP("E",E),C)); D(circumcircle(B,MP("F",F),D)); D(circumcircle(A,F,E)); [/asy]

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See also

Mock AIME 1 2007-2008 (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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