# Difference between revisions of "2012 USAMO Problems/Problem 5"

## Problem

Let $P$ be a point in the plane of triangle $ABC$, and $\gamma$ a line passing through $P$. Let $A'$, $B'$, $C'$ be the points where the reflections of lines $PA$, $PB$, $PC$ with respect to $\gamma$ intersect lines $BC$, $AC$, $AB$, respectively. Prove that $A'$, $B'$, $C'$ are collinear.

## Solution

By the sine law on triangle $AB'P$, $$\frac{AB'}{\sin \angle APB'} = \frac{AP}{\sin \angle AB'P},$$ so $$AB' = AP \cdot \frac{\sin \angle APB'}{\sin \angle AB'P}.$$ $[asy] import graph; import geometry; unitsize(0.5 cm); pair[] A, B, C; pair P, R; A = (2,12); B = (0,0); C = (14,0); P = (4,5); R = 5*dir(70); A = extension(B,C,P,reflect(P + R,P - R)*(A)); B = extension(C,A,P,reflect(P + R,P - R)*(B)); C = extension(A,B,P,reflect(P + R,P - R)*(C)); draw((P - R)--(P + R),red); draw(A--B--C--cycle,blue); draw(A--B--C--cycle); draw(A--P); draw(B--P); draw(C--P); draw(P--A); draw(P--B); draw(P--C); draw(A--B); draw(A--B); label("A", A, N); label("B", B, S); label("C", C, SE); dot("A'", A, SW); dot("B'", B, NE); dot("C'", C, W); dot("P", P, SE); label("\gamma", P + R, N); [/asy]$

Similarly, \begin{align*} B'C &= CP \cdot \frac{\sin \angle CPB'}{\sin \angle CB'P}, \\ CA' &= CP \cdot \frac{\sin \angle CPA'}{\sin \angle CA'P}, \\ A'B &= BP \cdot \frac{\sin \angle BPA'}{\sin \angle BA'P}, \\ BC' &= BP \cdot \frac{\sin \angle BPC'}{\sin \angle BC'P}, \\ C'A &= AP \cdot \frac{\sin \angle APC'}{\sin \angle AC'P}. \end{align*} Hence, \begin{align*} &\frac{AB'}{B'C} \cdot \frac{CA'}{A'B} \cdot \frac{BC'}{C'A} \\ &= \frac{\sin \angle APB'}{\sin \angle AB'P} \cdot \frac{\sin \angle CB'P}{\sin \angle CPB'} \cdot \frac{\sin \angle CPA'}{\sin \angle CA'P} \cdot \frac{\sin \angle BA'P}{\sin \angle BPA'} \cdot \frac{\sin \angle BPC'}{\sin \angle BC'P} \cdot \frac{\sin \angle AC'P}{\sin \angle APC'}. \end{align*}

Since angles $\angle AB'P$ and $\angle CB'P$ are supplementary or equal, depending on the position of $B'$ on $AC$, $$\sin \angle AB'P = \sin \angle CB'P.$$ Similarly, \begin{align*} \sin \angle CA'P &= \sin \angle BA'P, \\ \sin \angle BC'P &= \sin \angle AC'P. \end{align*}

By the reflective property, $\angle APB'$ and $\angle BPA'$ are supplementary or equal, so $$\sin \angle APB' = \sin \angle BPA'.$$ Similarly, \begin{align*} \sin \angle CPA' &= \sin \angle APC', \\ \sin \angle BPC' &= \sin \angle CPB'. \end{align*} Therefore, $$\frac{AB'}{B'C} \cdot \frac{CA'}{A'B} \cdot \frac{BC'}{C'A} = 1,$$ so by Menelaus's theorem, $A'$, $B'$, and $C'$ are collinear.