Difference between revisions of "2018 USAMO Problems/Problem 5"
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==Solution== | ==Solution== | ||
− | + | [asy] | |
+ | /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ | ||
+ | import graph; size(13cm); | ||
+ | real labelscalefactor = 0.5; /* changes label-to-point distance */ | ||
+ | pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ | ||
+ | pen dotstyle = black; /* point style */ | ||
+ | real xmin = -14.573333333333343, xmax = 3.56, ymin = -4.74, ymax = 8.473333333333338; /* image dimensions */ | ||
+ | pen sexdts = rgb(0.1803921568627451,0.49019607843137253,0.19607843137254902); pen cczzff = rgb(0.8,0.6,1); pen ccwwff = rgb(0.8,0.4,1); pen wvvxds = rgb(0.396078431372549,0.3411764705882353,0.8235294117647058); pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); | ||
+ | /* draw figures */ | ||
+ | draw(circle((-4.80390015600624,-0.5952574102964114), 2.6896620042551294), linewidth(1) + sexdts); | ||
+ | draw((-5.58,1.98)--(-4.06,-3.18), linewidth(1) + cczzff); | ||
+ | draw((-3.8846455730896308,1.9324397054436309)--(-7.42,-1.22), linewidth(1) + ccwwff); | ||
+ | draw(circle((-7.256640463424001,0.8150682664688008), 2.0416143581437347), linewidth(1) + sexdts); | ||
+ | draw(circle((-4.74356842508035,1.555353052957629), 0.9380526686465809), linewidth(1) + sexdts); | ||
+ | draw((-13.275816730447621,2.195893092761112)--(-3.912501589632712,1.120300499610314), linewidth(1) + wvvxds); | ||
+ | draw((-3.775365275873233,5.118495172394378)--(-8.947688716232484,-0.3288482488643849), linewidth(1) + wvvxds); | ||
+ | draw((-5.58,1.98)--(-3.775365275873233,5.118495172394378), linewidth(1) + wvvxds); | ||
+ | draw((-5.58,1.98)--(-7.42,-1.22), linewidth(1) + wvvxds); | ||
+ | draw((-3.8846455730896308,1.9324397054436309)--(-4.06,-3.18), linewidth(1) + rvwvcq); | ||
+ | draw((-3.8846455730896308,1.9324397054436309)--(-3.775365275873233,5.118495172394378), linewidth(1) + rvwvcq); | ||
+ | draw((-13.275816730447621,2.195893092761112)--(-4.06,-3.18), linewidth(1) + rvwvcq); | ||
+ | draw((-13.275816730447621,2.195893092761112)--(-3.775365275873233,5.118495172394378), linewidth(1) + rvwvcq); | ||
+ | draw((-8.947688716232484,-0.3288482488643849)--(-0.2874232022466262,3.539053630345969), linewidth(1) + rvwvcq); | ||
+ | draw((-7.42,-1.22)--(-0.2874232022466262,3.539053630345969), linewidth(1) + rvwvcq); | ||
+ | draw((-0.2874232022466262,3.539053630345969)--(-4.06,-3.18), linewidth(1) + rvwvcq); | ||
+ | draw(circle((-6.783982903277441,1.6253383695771872), 2.915556239332651), linewidth(1) + rvwvcq); | ||
+ | draw((-5.58,1.98)--(-3.8846455730896308,1.9324397054436309), linewidth(1) + ccwwff); | ||
+ | draw((-5.216225985226909,0.7450829498492438)--(-3.912501589632712,1.120300499610314), linewidth(1) + ccwwff); | ||
+ | draw((-8.947688716232484,-0.3288482488643849)--(-5.58,1.98), linewidth(1) + ccwwff); | ||
+ | draw((-8.947688716232484,-0.3288482488643849)--(-5.216225985226909,0.7450829498492438), linewidth(1) + ccwwff); | ||
+ | draw((-5.58,1.98)--(-3.912501589632712,1.120300499610314), linewidth(1) + ccwwff); | ||
+ | /* dots and labels */ | ||
+ | dot((-5.58,1.98),dotstyle); | ||
+ | label("<math>A</math>", (-5.52,2.113333333333337), NE * labelscalefactor); | ||
+ | dot((-7.42,-1.22),dotstyle); | ||
+ | label("<math>B</math>", (-7.36,-1.0866666666666638), NE * labelscalefactor); | ||
+ | dot((-4.06,-3.18),dotstyle); | ||
+ | label("<math>C</math>", (-4,-3.046666666666664), NE * labelscalefactor); | ||
+ | dot((-3.8846455730896308,1.9324397054436309),dotstyle); | ||
+ | label("<math>D</math>", (-3.8266666666666733,2.06), NE * labelscalefactor); | ||
+ | dot((-5.216225985226909,0.7450829498492438),linewidth(4pt) + dotstyle); | ||
+ | label("<math>E</math>", (-5.16,0.8466666666666699), NE * labelscalefactor); | ||
+ | dot((-3.775365275873233,5.118495172394378),linewidth(4pt) + dotstyle); | ||
+ | label("<math>F</math>", (-3.72,5.22), NE * labelscalefactor); | ||
+ | dot((-13.275816730447621,2.195893092761112),linewidth(4pt) + dotstyle); | ||
+ | label("<math>G</math>", (-13.226666666666675,2.3), NE * labelscalefactor); | ||
+ | dot((-8.947688716232484,-0.3288482488643849),linewidth(4pt) + dotstyle); | ||
+ | label("<math>P</math>", (-8.893333333333342,-0.22), NE * labelscalefactor); | ||
+ | dot((-3.912501589632712,1.120300499610314),linewidth(4pt) + dotstyle); | ||
+ | label("<math>Q</math>", (-3.88,1.18), NE * labelscalefactor); | ||
+ | dot((-0.2874232022466262,3.539053630345969),linewidth(4pt) + dotstyle); | ||
+ | label("<math>H</math>", (-0.24,3.6466666666666705), NE * labelscalefactor); | ||
+ | dot((-7.211833579631486,1.4993048370077748),linewidth(4pt) + dotstyle); | ||
+ | label("<math>M</math>", (-7.16,1.60666666666667), NE * labelscalefactor); | ||
+ | |||
+ | |||
+ | clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); | ||
+ | /* end of picture */ | ||
+ | [/asy] | ||
+ | ---------- | ||
+ | <cmath>\angle DEQ + \angle AED + \angle AEP = \angle DAQ + \angle AQD + \angle AEP = 180 - \angle ADC + \angle AEP = 180 - \angle ADC + \angle ABP = \angle ABP + \angle ABC = 180</cmath> | ||
+ | so <math>P,E,Q</math> are collinear. Furthermore, note that <math>DQBP</math> is cyclic because: | ||
+ | <cmath>\angle EDQ = \angle BAE = BPE.</cmath> | ||
+ | Notice that since <math>A</math> is the intersection of <math>(EDQ)</math> and <math>(BPE)</math>, it is the Miquel point of <math>DQBP</math>. It follows that <math>A,M,H</math> are collinear. It’s a well known property of Miquel points that <math>\angle EAH = 90</math>, so it follows that <math>MA \perp AE</math>, as desired. <math>\blacksquare</math> | ||
+ | ~AopsUser101 |
Revision as of 21:13, 14 June 2020
Problem 5
In convex cyclic quadrilateral we know that lines and intersect at lines and intersect at and lines and intersect at Suppose that the circumcircle of intersects line at and , and the circumcircle of intersects line at and , where and are collinear in that order. Prove that if lines and intersect at , then
Solution
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(13cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -14.573333333333343, xmax = 3.56, ymin = -4.74, ymax = 8.473333333333338; /* image dimensions */ pen sexdts = rgb(0.1803921568627451,0.49019607843137253,0.19607843137254902); pen cczzff = rgb(0.8,0.6,1); pen ccwwff = rgb(0.8,0.4,1); pen wvvxds = rgb(0.396078431372549,0.3411764705882353,0.8235294117647058); pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451);
/* draw figures */
draw(circle((-4.80390015600624,-0.5952574102964114), 2.6896620042551294), linewidth(1) + sexdts); draw((-5.58,1.98)--(-4.06,-3.18), linewidth(1) + cczzff); draw((-3.8846455730896308,1.9324397054436309)--(-7.42,-1.22), linewidth(1) + ccwwff); draw(circle((-7.256640463424001,0.8150682664688008), 2.0416143581437347), linewidth(1) + sexdts); draw(circle((-4.74356842508035,1.555353052957629), 0.9380526686465809), linewidth(1) + sexdts); draw((-13.275816730447621,2.195893092761112)--(-3.912501589632712,1.120300499610314), linewidth(1) + wvvxds); draw((-3.775365275873233,5.118495172394378)--(-8.947688716232484,-0.3288482488643849), linewidth(1) + wvvxds); draw((-5.58,1.98)--(-3.775365275873233,5.118495172394378), linewidth(1) + wvvxds); draw((-5.58,1.98)--(-7.42,-1.22), linewidth(1) + wvvxds); draw((-3.8846455730896308,1.9324397054436309)--(-4.06,-3.18), linewidth(1) + rvwvcq); draw((-3.8846455730896308,1.9324397054436309)--(-3.775365275873233,5.118495172394378), linewidth(1) + rvwvcq); draw((-13.275816730447621,2.195893092761112)--(-4.06,-3.18), linewidth(1) + rvwvcq); draw((-13.275816730447621,2.195893092761112)--(-3.775365275873233,5.118495172394378), linewidth(1) + rvwvcq); draw((-8.947688716232484,-0.3288482488643849)--(-0.2874232022466262,3.539053630345969), linewidth(1) + rvwvcq); draw((-7.42,-1.22)--(-0.2874232022466262,3.539053630345969), linewidth(1) + rvwvcq); draw((-0.2874232022466262,3.539053630345969)--(-4.06,-3.18), linewidth(1) + rvwvcq); draw(circle((-6.783982903277441,1.6253383695771872), 2.915556239332651), linewidth(1) + rvwvcq); draw((-5.58,1.98)--(-3.8846455730896308,1.9324397054436309), linewidth(1) + ccwwff); draw((-5.216225985226909,0.7450829498492438)--(-3.912501589632712,1.120300499610314), linewidth(1) + ccwwff); draw((-8.947688716232484,-0.3288482488643849)--(-5.58,1.98), linewidth(1) + ccwwff); draw((-8.947688716232484,-0.3288482488643849)--(-5.216225985226909,0.7450829498492438), linewidth(1) + ccwwff); draw((-5.58,1.98)--(-3.912501589632712,1.120300499610314), linewidth(1) + ccwwff);
/* dots and labels */
dot((-5.58,1.98),dotstyle); label("", (-5.52,2.113333333333337), NE * labelscalefactor); dot((-7.42,-1.22),dotstyle); label("", (-7.36,-1.0866666666666638), NE * labelscalefactor); dot((-4.06,-3.18),dotstyle); label("", (-4,-3.046666666666664), NE * labelscalefactor); dot((-3.8846455730896308,1.9324397054436309),dotstyle); label("", (-3.8266666666666733,2.06), NE * labelscalefactor); dot((-5.216225985226909,0.7450829498492438),linewidth(4pt) + dotstyle); label("", (-5.16,0.8466666666666699), NE * labelscalefactor); dot((-3.775365275873233,5.118495172394378),linewidth(4pt) + dotstyle); label("", (-3.72,5.22), NE * labelscalefactor); dot((-13.275816730447621,2.195893092761112),linewidth(4pt) + dotstyle); label("", (-13.226666666666675,2.3), NE * labelscalefactor); dot((-8.947688716232484,-0.3288482488643849),linewidth(4pt) + dotstyle); label("", (-8.893333333333342,-0.22), NE * labelscalefactor); dot((-3.912501589632712,1.120300499610314),linewidth(4pt) + dotstyle); label("", (-3.88,1.18), NE * labelscalefactor); dot((-0.2874232022466262,3.539053630345969),linewidth(4pt) + dotstyle); label("", (-0.24,3.6466666666666705), NE * labelscalefactor); dot((-7.211833579631486,1.4993048370077748),linewidth(4pt) + dotstyle); label("", (-7.16,1.60666666666667), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
so are collinear. Furthermore, note that is cyclic because: Notice that since is the intersection of and , it is the Miquel point of . It follows that are collinear. It’s a well known property of Miquel points that , so it follows that , as desired. ~AopsUser101