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2009 AIME I Problems/Problem 15

Revision as of 10:19, 9 August 2019 by Professor-mom (talk | contribs) (→‎Solution 3)

Problem

In triangle $ABC$, $AB = 10$, $BC = 14$, and $CA = 16$. Let $D$ be a point in the interior of $\overline{BC}$. Let $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD$, respectively. The circumcircles of triangles $BI_BD$ and $CI_CD$ meet at distinct points $P$ and $D$. The maximum possible area of $\triangle BPC$ can be expressed in the form $a - b\sqrt {c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.

Solution 1

First, by Law of Cosines, we have \[\cos BAC = \frac {16^2 + 10^2 - 14^2}{2\cdot 10 \cdot 16} = \frac {256+100-196}{320} = \frac {1}{2},\] so $\angle BAC = 60^\circ$.

Let $O_1$ and $O_2$ be the circumcenters of triangles $BI_BD$ and $CI_CD$, respectively. We first compute \[\angle BO_1D = \angle BO_1I_B + \angle I_BO_1D = 2\angle BDI_B + 2\angle I_BBD.\] Because $\angle BDI_B$ and $\angle I_BBD$ are half of $\angle BDA$ and $\angle ABD$, respectively, the above expression can be simplified to \[\angle BO_1D = \angle BO_1I_B + \angle I_BO_1D = 2\angle BDI_B + 2\angle I_BBD = \angle ABD + \angle BDA.\] Similarly, $\angle CO_2D = \angle ACD + \angle CDA$. As a result \begin{align*}\angle CPB &= \angle CPD + \angle BPD \\&= \frac {1}{2} \cdot \angle CO_2D + \frac {1}{2} \cdot \angle BO_1D \\&= \frac {1}{2}(\angle ABD + \angle BDA + \angle ACD + \angle CDA) \\&= \frac {1}{2} (2 \cdot 180^\circ - \angle BAC) \\&= \frac {1}{2} \cdot 300^\circ = 150^\circ.\end{align*}

Therefore $\angle CPB$ is constant ($150^\circ$). Also, $P$ is $B$ or $C$ when $D$ is $B$ or $C$. Let point $L$ be on the same side of $\overline{BC}$ as $A$ with $\overline{LC} = \overline{LB} = \overline {BC} = 14$; $P$ is on the circle with $L$ as the center and $\overline{LC}$ as the radius, which is $14$. The shortest distance from $L$ to $\overline{BC}$ is $7\sqrt {3}$.

When the area of $\triangle BPC$ is the maximum, the distance from $P$ to $\overline{BC}$ has to be the greatest. In this case, it's $14 - 7\sqrt {3}$. The maximum area of $\triangle BPC$ is \[\frac {1}{2} \cdot 14 \cdot (14 - 7\sqrt {3}) = 98 - 49 \sqrt {3}\] and the requested answer is $98 + 49 + 3 = \boxed{150}$.

Solution 2

From Law of Cosines on $\triangle{ABC}$, \[\cos{A}=\frac{16^2+10^2-14^2}{2\cdot 10\cdot 16}=\frac{1}{2}\implies\angle{A}=60^\circ.\]Now, \[\angle{CI_CD}+\angle{BI_BD}=180^\circ+\angle{\frac{A}{2}}=210^\circ.\]Since $CI_CDP$ and $BI_BDP$ are cyclic quadrilaterals, it follows that \[\angle{BPC}=\angle{CPD}+\angle{DPB}=(180^\circ-\angle{CI_CD})+(180^\circ-\angle{BI_BD})=360^\circ-210^\circ=150^\circ.\]Next, applying Law of Cosines on $\triangle{CPB}$, \begin{align*} & BC^2=14^2=PC^2+PB^2+2\cdot PB\cdot PC\cdot\frac{\sqrt{3}}{2} \\ & \implies \frac{PC^2+PB^2-196}{PC\cdot PB}=-\sqrt{3} \\ & \implies \frac{PC}{PB}+\frac{PB}{PC}-\frac{196}{PC\cdot PB}=-\sqrt{3} \\ & \implies PC\cdot PB = 196\left(\frac{1}{\frac{PC}{PB}+\frac{PB}{PC}+\sqrt{3}}\right). \end{align*} By AM-GM, $\frac{PC}{PB}+\frac{PB}{PC}\geq{2}$, so \[PB\cdot PC\leq 196\left(\frac{1}{2+\sqrt{3}}\right)=196(2-\sqrt{3}).\]Finally, \[[\triangle{BPC}]=\frac12 \cdot PB\cdot PC\cdot\sin{150^\circ}=\frac14 \cdot PB\cdot PC,\]and the maximum area would be $49(2-\sqrt{3})=98-49\sqrt{3},$ so the answer is $\boxed{150}$.

Solution 3

First, we notice that triangle ABC is a scaled version of a 5-7-8 triangle (which has a 60 degree angle opposite the side with length 7). So $\angle{BAC} = 60^\circ{}$. Therefore, let $\angle{I_B AB} = \angle{I_B AD} = \alpha$ and $\angle{I_C AD} = \angle{I_C AC} = 30 - \alpha.$ Therefore, in triangle $ABD$, we know that $\angle{BI_BD} = 90^\circ{} + \frac{\angle{BAD}}{2} = 90 + \alpha$ and $\angle{CI_CD} = 90^\circ{}+ \frac{\angle{CAD}}{2} = 90^\circ{} + (30^\circ{} - \alpha) = 120^\circ{} - \alpha$. Now note that quadrilaterals $BI_BDP$ and $CI_CDP$ are both cyclic. This means that $\angle{BPD} = 180^\circ{} - \angle{BI_BD} = 90^\circ{} + \alpha$ and $\angle{DPC} = 180^\circ{} - \angle{DI_CC} = 60^\circ + \alpha$. Therefore, $\angle{BPC} = 150^\circ{}$.

Now note that in order to maximize the area of $BPC$, we have to maximize the distance from $P$ to line $BC$. Note that since the second intersection of the Circles must lie below line $BC$, we try to find the locus of all points P under BC such that $\angle{BPC} = 150^\circ{}$. Let the circumcenter of triangle $BPC$ be $O$. Then major $\arc{BC} = 2 * \angle{BCP} = 300^\circ{}$ (Error compiling LaTeX. Unknown error_msg), minor $\arc{BC} = 60^\circ{}$ (Error compiling LaTeX. Unknown error_msg) which means $OBC$ is an equilateral triangle. (I will work on finishing solution maybe sometime later today...)

See also

2009 AIME I (Problems • Answer Key • Resources)
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