Difference between revisions of "2018 USAJMO Problems/Problem 3"

(Created page with "== Problem == (<math>*</math>) Let <math>ABCD</math> be a quadrilateral inscribed in circle <math>\omega</math> with <math>\overline{AC} \perp \overline{BD}</math>. Let <math>...")
 
Line 4: Line 4:
 
<asy>
 
<asy>
 
unitsize(3cm);  
 
unitsize(3cm);  
real labelscalefactor = 0.5; /* changes label-to-point distance */
+
real labelscalefactor = 1.5; /* changes label-to-point distance */
 
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */  
 
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */  
 
pen dotstyle = black; /* point style */  
 
pen dotstyle = black; /* point style */  
Line 30: Line 30:
 
/* dots and labels */
 
/* dots and labels */
 
dot((0,0),dotstyle);  
 
dot((0,0),dotstyle);  
label("$O$", (0.03388760411534265,0.08889671794036069), NE * labelscalefactor);  
+
label("$O$", (0.03388760411534265,0.08889671794036069), NE * 0.5);  
 
dot((0.8660254037844386,-0.5),dotstyle);  
 
dot((0.8660254037844386,-0.5),dotstyle);  
label("$B$", (0.9043736119372844,-0.4159851665963708), NE * labelscalefactor);  
+
label("$B$", (0.9043736119372844,-0.4159851665963708), S * 3.5 * labelscalefactor);  
 
dot((-0.9987724554622847,0.04953364724950819),dotstyle);  
 
dot((-0.9987724554622847,0.04953364724950819),dotstyle);  
label("$D$", (-0.9671713048798903,0.13242101833145825), NE * labelscalefactor);  
+
label("$D$", (-0.9671713048798903,0.13242101833145825), W * 3.5 * labelscalefactor);  
 
dot((0.5922362871684147,0.8057643453026269),dotstyle);  
 
dot((0.5922362871684147,0.8057643453026269),dotstyle);  
 
label("$A$", (0.625818089434263,0.8897438451365556), NE * labelscalefactor);  
 
label("$A$", (0.625818089434263,0.8897438451365556), NE * labelscalefactor);  
 
dot((0.06064145095757084,-0.9981596137020172),linewidth(4pt) + dotstyle);  
 
dot((0.06064145095757084,-0.9981596137020172),linewidth(4pt) + dotstyle);  
label("$C$", (0.09482162466287856,-0.9295719112113219), NE * labelscalefactor);  
+
label("$C$", (0.09482162466287856,-0.9295719112113219), S * 3);  
 
dot((2.3530139989292476,0.7523271151020314),dotstyle);  
 
dot((2.3530139989292476,0.7523271151020314),dotstyle);  
 
label("$E$", (2.3841998252345853,0.8375146846672384), NE * labelscalefactor);  
 
label("$E$", (2.3841998252345853,0.8375146846672384), NE * labelscalefactor);  
 
dot((0.5249726058304045,-2.4139334560841545),dotstyle);  
 
dot((0.5249726058304045,-2.4139334560841545),dotstyle);  
label("$F$", (0.5561792088085077,-2.3310543838046627), NE * labelscalefactor);  
+
label("$F$", (0.5561792088085077,-2.3310543838046627), S * 3.5 * labelscalefactor);  
 
dot((1.5189031419104242,-0.6923952683993904),linewidth(4pt) + dotstyle);  
 
dot((1.5189031419104242,-0.6923952683993904),linewidth(4pt) + dotstyle);  
label("$P$", (1.5572381178037407,-0.6249018084736391), NE * labelscalefactor);  
+
label("$P$", (1.5572381178037407,-0.6249018084736391), SE * 3);  
 
dot((0.3454211217688861,-0.9384477868458769),linewidth(4pt) + dotstyle);  
 
dot((0.3454211217688861,-0.9384477868458769),linewidth(4pt) + dotstyle);  
label("$Q$", (0.38208200724411934,-0.8686378906637853), NE * labelscalefactor);  
+
label("$Q$", (0.38208200724411934,-0.8686378906637853), S * 3);  
 
dot((0.9854301844182564,0.17008042696736536),linewidth(4pt) + dotstyle);  
 
dot((0.9854301844182564,0.17008042696736536),linewidth(4pt) + dotstyle);  
 
label("$R$", (1.0175367929541368,0.23687933927009236), NE * labelscalefactor);
 
label("$R$", (1.0175367929541368,0.23687933927009236), NE * labelscalefactor);
Line 52: Line 52:
  
 
==Solution 1==
 
==Solution 1==
 +
First we have that <math>BE=BD=BF</math> by the definition of a reflection. Let <math>\angle DEB = \alpha</math> and <math>\angle DFB = \beta.</math> Since <math>\triangle DBE.</math> is isosceles we have <math>\angle BDE = \alpha.</math> Also, we see that <math>\angle BDE = \angle CAB = \angle CDB = \alpha,</math> using similar triangles and the property of cyclic quadrilaterals. Similarly, <cmath>\angle DFB = \angle FDB = \angle ACB = \angle ADB = \beta.</cmath> Now, from <math>BE=BD=BF</math> we know that <math>B</math> is the circumcenter of <math>\triangle DEF.</math> Using the properties of the circumcenter and some elementary angle chasing, we find that <cmath>\angle DPE = 90^{\circ} + \beta - \alpha.</cmath>
 +
 +
Now, we claim that <math>Q</math> is the intersection of ray <math>\overrightarrow{EB}</math> and the circumcircle of <math>ABCD.</math> To prove this, we just need to show that <math>DEPQ</math> is cyclic by this definition of <math>Q.</math> We have that <cmath>\angle DQE = \angle DCB = \angle DCA + \angle ACB = (90^{\circ}-\alpha)+\beta.</cmath> We also have from before that <cmath>\angle DPE = 90+\beta-\alpha,</cmath> so <math>\angle DQE=\angle DPE</math> and this proves the claim.
 +
 +
We can use a similar proof to show that <math>F, B, R</math> are collinear.
 +
 +
Now, <math>DP</math> is the radical axis of the circumcircles of <math>\triangle EDP</math> and <math>\triangle FDP.</math> Since <math>B</math> lies on <math>DP,</math> and <math>E, Q</math> lie on the circumcircle of <math>\triangle EPD</math> and <math>F, R</math> lie on the circumcircle of <math>\triangle FPD,</math> we have that <cmath>BF \cdot BR = BE \cdot BQ.</cmath> However, <math>BF=BE,</math> so <math>BR=BQ.</math> Since <math>E, B, Q</math> are collinear and so are <math>F, B, R</math> we can add these <math>2</math> equations to get <cmath>EQ=BE+BQ=BF+BR=FR,</cmath> which completes the proof.
  
  

Revision as of 11:48, 20 April 2018

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.


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

See also

2018 USAJMO (ProblemsResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6
All USAJMO Problems and Solutions
Invalid username
Login to AoPS