AIME 2007 II · 第 15 题

Solution 1 (Homothety)

First, apply Heron's formula to find that $[ABC] = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = 84$. 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 $O$. Note that $\triangle O_A O_B O_C$ and $\triangle ABC$ are similar because they have parallel sides and have the same incenter. Also, since the inradius of $\triangle O_A O_B O_C$ is $4-r$ and the inradius of $\triangle ABC$ is $4,$ it follows that $\triangle O_A O_B O_C$ is the result of a homothety $\mathcal{H}\left(I, \frac{4-r}{4}\right)$ on $\triangle ABC.$ Now, note that $OO_A = OO_B = OO_C = 2r$ are all circumradii of $\triangle O_A O_B O_C$, and the circumradius of $\triangle ABC$ is $R = \frac{abc}{4[ABC]} = \frac{65}{8}.$ From the ratio of the homothety, we have that $\frac{2r}{R}=\frac{4-r}{4}.$ Plugging in $R=\frac{65}{8}$ and solving yields $r = \frac{260}{129}$ and the final answer is $\boxed{389}$.

https://latex.artofproblemsolving.com/9/4/7/947b7f06d947dbf8bc5d8f61cdd193c330377372.png

Solution 2

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 $260+129=\boxed{389}$.

Solution 3 (elementary)

Let $A'$, $B'$, $C'$, and $O$ be the centers of circles $\omega_{A}$, $\omega_{B}$, $\omega_{C}$, $\omega$, respectively, and let $x$ be their radius.

Now, triangles $ABC$ and $A'B'C'$ are similar by parallel sides, so we can find ratios of two quantities in each triangle and set them equal to solve for $x$.

Since $OA'=OB'=OC'=2x$, $O$ is the circumcenter of triangle $A'B'C'$ and its circumradius is $2x$. Let $I$ denote the incenter of triangle $ABC$ and $r$ the inradius of $ABC$. Then the inradius of $A'B'C'=r-x$, so now we compute r. Computing the inradius by $A=rs$, we find that the inradius of $ABC$ is $4$. Additionally, using the circumradius formula $R=\frac{abc}{4K}$ where $K$ is the area of $ABC$ and $R$ is the circumradius, we find $R=\frac{65}{8}$. Now we can equate the ratio of circumradius to inradius in triangles $ABC$ and $A'B'C'$.

$$\frac{\frac{65}{8}}{4}=\frac{2x}{4-x}$$ Solving, we get $x=\frac{260}{129}$, so our answer is $260+129=\boxed{389}$.

Solution 4

According to the diagram, it is easily to see that there is a small triangle made by the center of three circles which aren't in the middle. The circumradius of them is$2r$. Now denoting $AB=13;BC=14;AC=15$, and centers of circles tangent to $AB,AC;AC,BC;AB,BC$ are relatively $M,N,O$ with $OJ,NK$ both perpendicular to $BC$. It is easy to know that $tanB=\frac{12}{5}$, so $tan\angle OBJ=\frac{2}{3}$ according to half angle formula. Similarly, we can find $tan\angle NCK=\frac{1}{2}$. So we can see that $JK=ON=14-\frac{7x}{2}$. Obviously, $\frac{2x}{14-\frac{7x}{2}}=\frac{65}{112}$ . After solving, we get $x=\frac{260}{129}$, so our answer is $260+129=\boxed{389}$. ~bluesoul

Sidenote (Generalization)

If 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 $ABC$ has side lengths $a,b,$ and $c$, then the radius of $\omega$ can be written as

$$\frac{abc\sqrt{2b^{2}c^{2}+2a^{2}b^{2}+2a^{2}c^{2}-a^{4}-b^{4}-c^{4}}}{4b^{2}c^{2}+4a^{2}b^{2}+4a^{2}c^{2}-2a^{4}-2b^{4}-2c^{4}+2a^{2}bc+2ab^{2}c+2abc^{2}},$$ or, more simply as,

$$\frac{abc\cdot K}{8K^{2}+sabc},$$ where $K$ is the area of the triangle and $s$ is the semiperimeter

~pinkpig

Solution 5

Let $O_A, O_B, O_C, O$ be the centers of $w_A, w_B, w_C, w$, respectively. Also, let $I$ be the incenter of $ABC$ and $r$ be the radius of circle $w$. Since $AB||O_AO_B$, $BC||O_BO_C$, and $CA||O_CO_A$, we know that

$$\angle BAI = \angle O_BO_AI, \angle CBI = \angle O_CO_BI, \angle ACI = \angle O_AO_CI \text{ and }\angle CAI = \angle O_CO_AI, \angle ABI = \angle O_AO_BI, \angle BCI = \angle O_BO_CI.$$ That means $\angle ABC = \angle O_AO_BO_C$, $\angle BAC = \angle O_BO_AO_C$, and $\angle ACB = \angle O_AO_CO_B$. Thus, $\triangle ABC \sim \triangle O_AO_BO_C$. We also know that we are scaling each side of $\triangle ABC$ (from $AB$ to $O_AO_B$ for instance), about $I$ (since A,O_A,I are collinear; same apply with $B$ and $C$).

Now, let the homothety $\mathcal{H} (I, x)$ map $\triangle ABC$ to $\triangle O_AO_BO_C$. To start off, we know the circumradius of $O_AO_BO_C$ is $O$, since $OO_A = OO_B = OO_C = 2r$. Since $O_AO_B = 13x$, $O_BO_C = 14x$, $O_CO_A = 15x$, we can get an relationship involving $x$ and $r$ via another way to find the circumradius:

$$[\triangle O_AO_BO_C ] =\frac{abc}{4R} \Longrightarrow 84x^2 =\frac{13x\cdot 14x\cdot 15x}{4\cdot 2r} \Longrightarrow r=\frac{65x}{16}$$ Take notice of the inradius of $ABC$. We get the inradius to be $[\triangle ABC ] = sr_0 \Longrightarrow r_0=4$. Let the tangency point of the incircle and side $BC$ be $H$. We know $IH = 4$. We also know that we can cut off the part of $IH$ that is outside of $\triangle O_AO_BO_C$ to get the inradius of $\triangle O_AO_BO_C$. To part that is outside $\triangle O_AO_BO_C$ turns out just to be the radius of circle $w_B$ (as seen in the picture). That means the inradius of $\triangle O_AO_BO_C$ is just $4-r$. We can calculate that incradius in another way, though. We know that the inradius of $\triangle ABC$ is $4$, which means the inradius of $\triangle O_AO_BO_C$ is just $4x$ (by our homethety ratio).

Thus, we have $4x = 4-r = 4-\dfrac{65x}{16} \Longrightarrow x = \dfrac{64}{129} \Longrightarrow r = \dfrac{260}{129}$. That gives $\boxed{389}$ as our final answer.

The homethety turned out to be $\mathcal{H} \left(I, \dfrac{64}{129}\right)$

~sml1809

Solution 6

Let the radius of $\omega$ be $r$, let the centers of $\omega_A$, $\omega_B$, and $\omega_C$ be $O_A$, $O_B$, and $O_C$, respectively, and let the feet of the altitudes from $O_A$ and $O_C$ onto $AC$ be $X$ and $Y$, respectively. Now, observe that $\triangle O_AO_BO_C$ has all its sides parallel to the sides of $\triangle ABC$, so the two triangles are similar. Then, we get \begin{align*} AC&=AX+XY+YC,\\ 15&=r\cot\frac{\angle A}{2}+O_AO_C+r\cot\frac{\angle C}{2},\\ 15&=r\sqrt{\frac{1+\cos A}{1-\cos A}}+4r\sin\angle B+r\sqrt{\frac{1+\cos C}{1-\cos C}},\\ 15&=r\sqrt{\frac{65+33}{65-33}}+\frac{48}{13}r+r\sqrt{\frac{5+3}{5-3}},\\ r&=\frac{15}{\frac 74+\frac{48}{13}+2}\\ &=\boxed{\frac{260}{129}}.\,\square\\ \end{align*}

~pieMax2713

Video Solution

2007 AIME II #15

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