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Difference between revisions of "2007 AIME II Problems/Problem 15"

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First, apply [[Heron's formula]] to find that the area is <math>\sqrt{21 \cdot 8 \cdot 7 \cdot 6} = 84</math>. Also the semiperimeter is <math>21</math>. So the [[inradius]] is <math>\frac{A}{s} = \frac{84}{21} = 4</math>.  
 
First, apply [[Heron's formula]] to find that the area is <math>\sqrt{21 \cdot 8 \cdot 7 \cdot 6} = 84</math>. Also the semiperimeter is <math>21</math>. So the [[inradius]] is <math>\frac{A}{s} = \frac{84}{21} = 4</math>.  
  
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Now consider the [[incenter]] I. Let the [[radius]] of one of the small circles be <math>r</math>. Let the centers of the three little circles tangent to the sides of <math>\triangle ABC</math> be <math>X</math>, <math>Y</math>, and <math>Z</math>. Let the centre of the circle tangent to those three circles be P. A [[homothety]] centered at <math>I</math> takes <math>XYZ</math> to <math>ABC</math> with factor <math>\frac{4 - r}{4}</math>. The same homothety takes <math>P</math> to the circumcentre of <math>\triangle ABC</math>, so <math>\frac{PX}R = \frac{2r}R = \frac{4 - r}4</math>, where <math>R</math> is the [[circumradius]] of <math>\triangle ABC</math>. The circumradius of <math>\triangle ABC</math> can be easily computed by <math>R = \frac a{2\sin A}</math>, so doing that reveals <math>R = \frac{65}8</math>. Then <math>\frac{2r}{\frac{65}{8}} = \frac{(4-r)}4 \Longrightarrow \frac{16r}{65} = \frac{4 - r}4 \Longrightarrow \frac{129r}{260} = 1 \Longrightarrow r = \frac{260}{129}</math>, so the answer is <math>\boxed{389}</math>.
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Now consider the [[incenter]] <math>I</math> of <math>\triangle ABC</math>. Let the [[radius]] of one of the small circles be <math>r</math>. Let the centers of the three little circles tangent to the sides of <math>\triangle ABC</math> be <math>O_A</math>, <math>O_B</math>, and <math>O_C</math>. Let the center of the circle tangent to those three circles be <math>O</math>. The [[homothety]] <math>\mathcal{H}(I, \frac{4-r}{4})</math> maps <math>\triangle ABC</math> to <math>\triangle XYZ</math>; since <math>OO_A = OO_B = OO_C = 2r</math>, <math>O</math> is the circumcenter of <math>\triangle XYZ</math> and <math>\mathcal{H}</math> therefore maps the circumcenter of <math>\triangle ABC</math> to <math>O</math>. Thus, <math>2r = R \cdot \frac{4 - r}{4}</math>, where <math>R</math> is the [[circumradius]] of <math>\triangle ABC</math>. Substituting <math>R = \frac{abc}{[ABC]} = \frac{65}{8}</math>, <math>r = \frac{260}{129}</math> and the answer is <math>\boxed{389}</math>.
  
 
=== Solution 2 ===
 
=== Solution 2 ===

Revision as of 17:16, 27 November 2020

Problem

Four circles $\omega,$ $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_{A}$ is tangent to sides $AB$ and $AC$, $\omega_{B}$ to $BC$ and $BA$, $\omega_{C}$ to $CA$ and $CB$, and $\omega$ is externally tangent to $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$. If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\omega$ can be represented in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

2007 AIME II-15.png

Solution 1

First, apply Heron's formula to find that the area is $\sqrt{21 \cdot 8 \cdot 7 \cdot 6} = 84$. Also the semiperimeter is $21$. So the inradius is $\frac{A}{s} = \frac{84}{21} = 4$.

Now consider the incenter $I$ of $\triangle ABC$. Let the radius of one of the small circles be $r$. Let the centers of the three little circles tangent to the sides of $\triangle ABC$ be $O_A$, $O_B$, and $O_C$. Let the center of the circle tangent to those three circles be $O$. The homothety $\mathcal{H}(I, \frac{4-r}{4})$ maps $\triangle ABC$ to $\triangle XYZ$; since $OO_A = OO_B = OO_C = 2r$, $O$ is the circumcenter of $\triangle XYZ$ and $\mathcal{H}$ therefore maps the circumcenter of $\triangle ABC$ to $O$. Thus, $2r = R \cdot \frac{4 - r}{4}$, where $R$ is the circumradius of $\triangle ABC$. Substituting $R = \frac{abc}{[ABC]} = \frac{65}{8}$, $r = \frac{260}{129}$ and the answer is $\boxed{389}$.

Solution 2

2007 AIME II-15b.gif

Consider a 13-14-15 triangle. $A=84.$ [By Heron's Formula or by 5-12-13 and 9-12-15 right triangles.]

The inradius is $r=\frac{A}{s}=\frac{84}{21}=4$, where $s$ is the semiperimeter. Scale the triangle with the inradius by a linear scale factor, $u.$

The circumradius is $R=\frac{abc}{4rs}=\frac{13\cdot 14\cdot 15}{4\cdot 4\cdot 21}=\frac{65}{8},$ where $a,$ $b,$ and $c$ are the side-lengths. Scale the triangle with the circumradius by a linear scale factor, $v$.

Cut and combine the triangles, as shown. Then solve for $4u$:

$\frac{65}{8}v=8u$
$v=\frac{64}{65}u$
$u+v=1$
$u+\frac{64}{65}u=1$
$\frac{129}{65}u=1$
$4u=\frac{260}{129}$

The solution is $260+129=\boxed{389}$.

See also

2007 AIME II (Problems • Answer Key • Resources)
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