Problem 53: Hard geometry problem

by henderson, Oct 9, 2016, 7:34 AM

${\color{red}\bf{Problem \ 53}}$ Let $ABC$ be a triangle and $E,F$ be two points on $AC$ and $AB,$ respectively, such that $EFBC$ is cyclic. $M$ is the midpoint of $BC,$ $AM\cap \odot(\triangle AEF)=D$ and $AM\cap \odot(\triangle ABC)=Q.$ $E',F'$ are the reflections of $A$ with respect to $E$ and $F,$ respectively. The tangent from $D$ to $\odot(\triangle AEF)$ cuts $BC$ at $P.$ If $E',F',P$ lie on a line, show that $PA=PQ.$

$[asy] import graph; size(15.239250143446121cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ real xmin = -7.034247808662772, xmax = 8.20500233478335, ymin = -8.34322678066502, ymax = 6.028482806816687; /* image dimensions */ pen uuuuuu = rgb(0.26666666666666666,0.26666666666666666,0.26666666666666666); /* draw figures */ draw(circle((1.6952295227991048,1.884525129494876), 1.4819072791149188)); draw(circle((2.8159616023769294,-0.8946244648443626), 4.422450463096355)); draw((1.626383577474811,3.3648323339987876)--(0.490794000592034,1.021168739155659)); draw((0.490794000592034,1.021168739155659)--(-0.644795576290743,-1.3224948556874696)); draw((1.626383577474811,3.3648323339987876)--(3.1727183204641243,1.7701746302910917)); draw((3.1727183204641243,1.7701746302910917)--(4.719053063453438,0.1755169265833958)); draw((2.886331467888337,-2.407698595874594)--(7.036808827903741,-2.2146687055059293)); draw((1.626383577474811,3.3648323339987876)--(2.2494615386213277,0.5101612560860114)); draw((2.2494615386213277,0.5101612560860114)--(2.886331467888337,-2.407698595874594)); draw((2.886331467888337,-2.407698595874594)--(3.509409429034852,-5.262369673787366)); draw((3.1727183204641243,1.7701746302910917)--(1.8317561605280792,1.3956716847233754)); draw((1.8317561605280792,1.3956716847233754)--(0.490794000592034,1.021168739155659)); draw((4.719053063453438,0.1755169265833958)--(7.036808827903741,-2.2146687055059293)); draw((-0.644795576290743,-1.3224948556874696)--(-1.2641458921270663,-2.6007284862432596)); draw((3.1727183204641243,1.7701746302910917)--(2.2494615386213277,0.5101612560860114)); draw((2.2494615386213277,0.5101612560860114)--(0.490794000592034,1.021168739155659)); draw((7.036808827903741,-2.2146687055059293)--(3.509409429034852,-5.262369673787366)); draw((4.719053063453438,0.1755169265833958)--(-6.012417843477453,-2.8215605339107523)); draw((-6.012417843477453,-2.8215605339107523)--(1.626383577474811,3.3648323339987876), red); draw((-6.012417843477453,-2.8215605339107523)--(-1.2641458921270663,-2.6007284862432596)); draw((-1.2641458921270663,-2.6007284862432596)--(2.886331467888337,-2.407698595874594)); draw((-6.012417843477453,-2.8215605339107523)--(3.509409429034852,-5.262369673787366), red); draw((2.2494615386213277,0.5101612560860114)--(-6.012417843477453,-2.8215605339107523)); draw(circle((2.884807547701221,-2.3749316693482756), 4.155093121505569)); /* dots and labels */ dot((1.626383577474811,3.3648323339987876)); label("$A$", (1.5920516817734325,3.687760174686978), NE * labelscalefactor); dot((0.490794000592034,1.021168739155659)); label("$F$", (0.0042887774616452825,1.003295058118641), NE * labelscalefactor); dot((3.1727183204641243,1.7701746302910917)); label("$E$", (3.3762388629072966,1.9363103730234899), NE * labelscalefactor); dot((-0.644795576290743,-1.3224948556874696)); label("$F'$", (-0.8468830887673541,-1.1410032971889938), NE * labelscalefactor); dot((4.719053063453438,0.1755169265833958)); label("$E'$", (4.833052249337699,0.39765353791724795), NE * labelscalefactor); dot((2.2494615386213277,0.5101612560860114),linewidth(3.pt) + uuuuuu); label("$D$", (2.4432235480024316,0.15212319188965615), NE * labelscalefactor,uuuuuu); dot((-6.012417843477453,-2.8215605339107523),linewidth(3.pt) + uuuuuu); label("$P$", (-6.428606288461369,-2.7615035809711), NE * labelscalefactor,uuuuuu); dot((-1.2641458921270663,-2.6007284862432596),linewidth(3.pt) + uuuuuu); label("$B$", (-1.6980549549963533,-3.0234026167338643), NE * labelscalefactor,uuuuuu); dot((7.036808827903741,-2.2146687055059293),linewidth(3.pt) + uuuuuu); label("$C$", (7.3210930890840045,-2.499604545208335), NE * labelscalefactor,uuuuuu); dot((3.509409429034852,-5.262369673787366),linewidth(3.pt) + uuuuuu); label("$Q$", (3.310764103966604,-5.724236423037374), NE * labelscalefactor,uuuuuu); dot((2.886331467888337,-2.407698595874594),linewidth(3.pt) + uuuuuu); label("$M$", (2.3777487890617395,-2.7615035809711), NE * labelscalefactor,uuuuuu); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]$