2011 USAMO Problems/Problem 5

Problem

Let $P$ be a given point inside quadrilateral $ABCD$ . Points $Q_1$ and $Q_2$ are located within $ABCD$ such that $\angle Q_1 BC = \angle ABP$ , $\angle Q_1 CB = \angle DCP$ , $\angle Q_2 AD = \angle BAP$ , $\angle Q_2 DA = \angle CDP$ . Prove that $\overline{Q_1 Q_2} \parallel \overline{AB}$ if and only if $\overline{Q_1 Q_2} \parallel \overline{CD}$ .

Solution

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First note that $\overline{Q_1 Q_2} \parallel \overline{AB}$ if and only if the altitudes from $Q_1$ and $Q_2$ to $\overline{AB}$ are the same, or $|Q_1B|\sin \angle ABQ_1 =|Q_2A|\sin \angle BAQ_2$ . Similarly $\overline{Q_1 Q_2} \parallel \overline{CD}$ iff $|Q_1C|\sin \angle DCQ_1 =|Q_2D|\sin \angle CDQ_2$ .

If we define $S =\frac{|Q_1B|\sin \angle ABQ_1}{|Q_2A|\sin \angle BAQ_2}\frac{|Q_2D|\sin \angle CDQ_2}{|Q_1C|\sin \angle DCQ_1}$ , then we are done if we can show that S=1.

By the law of sines, $\frac{|Q_1B|}{|Q_1C|}=\frac{\sin\angle Q_1CB}{\sin\angle Q_1BC}$ and $\frac{|Q_2D|}{|Q_2A|}=\frac{\sin\angle Q_2AD}{\sin\angle Q_2DA}$ .

So, $S=\frac{\sin \angle ABQ_1}{\sin \angle BAQ_2}\frac{\sin \angle CDQ_2}{\sin \angle DCQ_1}\frac{\sin \angle BCQ_1}{\sin \angle CBQ_1}\frac{\sin \angle DAQ_2}{\sin \angle ADQ_2}$

By the terms of the problem, $S=\frac{\sin \angle PBC}{\sin \angle PAD}\frac{\sin \angle PDA}{\sin \angle PCB}\frac{\sin \angle PCD}{\sin \angle PBA}\frac{\sin \angle PAB}{\sin \angle PDC} = \frac{\sin \angle PBC}{\sin \angle PCB}\frac{\sin \angle PDA}{\sin \angle PAD}\frac{\sin \angle PCD}{\sin \angle PDC}\frac{\sin \angle PAB}{\sin \angle PBA}$ .

Applying the law of sines to the triangles with vertices at P yields $S=\frac{|PC|}{|PB|}\frac{|PA|}{|PD|}\frac{|PD|}{|PC|}\frac{|PB|}{|PA|}=1$ .

2011 USAMO (Problems • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6
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2011 USAMO Problems/Problem 5

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