2018 USAJMO Problems/Problem 3

Problem

( $*$ ) Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$ . Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$ , respectively, and let $P$ be the intersection of lines $BD$ and $EF$ . Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$ , and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$ . Show that $EQ = FR$ .

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Solution 1

First we have that $BE=BD=BF$ by the definition of a reflection. Let $\angle DEB = \alpha$ and $\angle DFB = \beta.$ Since $\triangle DBE.$ is isosceles we have $\angle BDE = \alpha.$ Also, we see that $\angle BDE = \angle CAB = \angle CDB = \alpha,$ using similar triangles and the property of cyclic quadrilaterals. Similarly, $\angle DFB = \angle FDB = \angle ACB = \angle ADB = \beta.$ Now, from $BE=BD=BF$ we know that $B$ is the circumcenter of $\triangle DEF.$ Using the properties of the circumcenter and some elementary angle chasing, we find that $\angle DPE = 90^{\circ} + \beta - \alpha.$

Now, we claim that $Q$ is the intersection of ray $\overrightarrow{EB}$ and the circumcircle of $ABCD.$ To prove this, we just need to show that $DEPQ$ is cyclic by this definition of $Q.$ We have that $\angle DQE = \angle DCB = \angle DCA + \angle ACB = (90^{\circ}-\alpha)+\beta.$ We also have from before that $\angle DPE = 90+\beta-\alpha,$ so $\angle DQE=\angle DPE$ and this proves the claim.

We can use a similar proof to show that $F, B, R$ are collinear.

Now, $DP$ is the radical axis of the circumcircles of $\triangle EDP$ and $\triangle FDP.$ Since $B$ lies on $DP,$ and $E, Q$ lie on the circumcircle of $\triangle EPD$ and $F, R$ lie on the circumcircle of $\triangle FPD,$ we have that $BF \cdot BR = BE \cdot BQ.$ However, $BF=BE,$ so $BR=BQ.$ Since $E, B, Q$ are collinear and so are $F, B, R$ we can add these $2$ equations to get $EQ=BE+BQ=BF+BR=FR,$ which completes the proof.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

2018 USAJMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
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2018 USAJMO Problems/Problem 3

Problem

Solution 1

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