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 $EQ = FR$ .

$[asy] unitsize(3cm); real labelscalefactor = 1.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); draw((0.5922362871684147,0.8057643453026269)--(0.8660254037844386,-0.5)--(0.06064145095757084,-0.9981596137020172)--(-0.9987724554622847,0.04953364724950819)--cycle, linewidth(2) + rvwvcq); /* draw figures */ draw(circle((0,0), 1), linewidth(2) + wrwrwr); draw((0.8660254037844386,-0.5)--(-0.9987724554622847,0.04953364724950819), linewidth(2) + wrwrwr); draw((0.5922362871684147,0.8057643453026269)--(0.8660254037844386,-0.5), linewidth(2) + rvwvcq); draw((0.8660254037844386,-0.5)--(0.06064145095757084,-0.9981596137020172), linewidth(2) + rvwvcq); draw((0.06064145095757084,-0.9981596137020172)--(-0.9987724554622847,0.04953364724950819), linewidth(2) + rvwvcq); draw((-0.9987724554622847,0.04953364724950819)--(0.5922362871684147,0.8057643453026269), linewidth(2) + rvwvcq); draw((0.5922362871684147,0.8057643453026269)--(0.06064145095757084,-0.9981596137020172), linewidth(2) + wrwrwr); draw((0.5249726058304045,-2.4139334560841545)--(2.3530139989292476,0.7523271151020314), linewidth(2) + wrwrwr); draw((-0.9987724554622847,0.04953364724950819)--(1.5189031419104242,-0.6923952683993904), linewidth(2) + wrwrwr); draw((-0.9987724554622847,0.04953364724950819)--(0.5249726058304045,-2.4139334560841545), linewidth(2) + wrwrwr); draw((-0.9987724554622847,0.04953364724950819)--(2.3530139989292476,0.7523271151020314), linewidth(2) + wrwrwr); draw((0.8660254037844386,-0.5)--(2.3530139989292476,0.7523271151020314), linewidth(2) + wrwrwr); draw((0.8660254037844386,-0.5)--(0.5249726058304045,-2.4139334560841545), linewidth(2) + wrwrwr); draw(circle((0.5922362871684147,0.8057643453026269), 1.7615883990890795), linewidth(2) + linetype("4 4") + wrwrwr); draw(circle((0.06064145095757076,-0.9981596137020177), 1.4899728165839203), linewidth(2) + linetype("4 4") + wrwrwr); draw((0.5249726058304045,-2.4139334560841545)--(0.9854301844182564,0.17008042696736536), linewidth(2) + wrwrwr); draw((2.3530139989292476,0.7523271151020314)--(0.3454211217688861,-0.9384477868458769), linewidth(2) + wrwrwr); /* dots and labels */ dot((0,0),dotstyle); label("$O$", (0.03388760411534265,0.08889671794036069), NE * 0.5); dot((0.8660254037844386,-0.5),dotstyle); label("$B$", (0.9043736119372844,-0.4159851665963708), S * 3.5 * labelscalefactor); dot((-0.9987724554622847,0.04953364724950819),dotstyle); label("$D$", (-0.9671713048798903,0.13242101833145825), W * 3.5 * labelscalefactor); dot((0.5922362871684147,0.8057643453026269),dotstyle); label("$A$", (0.625818089434263,0.8897438451365556), NE * labelscalefactor); dot((0.06064145095757084,-0.9981596137020172),linewidth(4pt) + dotstyle); label("$C$", (0.09482162466287856,-0.9295719112113219), S * 3); dot((2.3530139989292476,0.7523271151020314),dotstyle); label("$E$", (2.3841998252345853,0.8375146846672384), NE * labelscalefactor); dot((0.5249726058304045,-2.4139334560841545),dotstyle); label("$F$", (0.5561792088085077,-2.3310543838046627), S * 3.5 * labelscalefactor); dot((1.5189031419104242,-0.6923952683993904),linewidth(4pt) + dotstyle); label("$P$", (1.5572381178037407,-0.6249018084736391), SE * 3); dot((0.3454211217688861,-0.9384477868458769),linewidth(4pt) + dotstyle); label("$Q$", (0.38208200724411934,-0.8686378906637853), S * 3); dot((0.9854301844182564,0.17008042696736536),linewidth(4pt) + dotstyle); label("$R$", (1.0175367929541368,0.23687933927009236), NE * labelscalefactor); [/asy]$

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.

~nukelauncher

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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