Problem

Source: Iran second round 2016 problem 2 Day 1

Tags: geometry, cyclic quadrilateral



Let $ABC$ be a triangle such that $\angle C=2\angle B$ and $\omega$ be its circumcircle. a tangent from $A$ to $\omega$ intersect $BC$ at $E$. $\Omega$ is a circle passing throw $B$ that is tangent to $AC$ at $C$. Let $\Omega\cap AB=F$. $K$ is a point on $\Omega$ such that $EK$ is tangent to $\Omega$ ($A,K$ aren't in one side of $BC$). Let $M$ be the midpoint of arc $BC$ of $\omega$ (not containing $A$). Prove that $AFMK$ is a cyclic quadrilateral. [asy][asy] import graph; size(15.424606256655986cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ real xmin = -7.905629294221492, xmax = 11.618976962434495, ymin = -5.154837585051625, ymax = 4.0091473316396895; /* image dimensions */ pen uuuuuu = rgb(0.26666666666666666,0.26666666666666666,0.26666666666666666); /* draw figures */ draw(circle((1.4210145017438194,0.18096629151696939), 2.581514123077079)); draw(circle((1.4210145017438194,-1.3302878964546825), 2.8984706754484924)); draw(circle((-0.7076932767793396,-0.4161825262831505), 2.9101722408015513), linetype("4 4") + red); draw((3.996177869179178,0.)--(-3.839514259733819,0.)); draw((3.996177869179178,0.)--(0.07833180472267817,2.385828723227042)); draw((0.07833180472267817,2.385828723227042)--(-1.154148865691539,0.)); draw((-3.839514259733819,0.)--(-0.6807342461448075,-3.3262298939043657)); draw((0.07833180472267817,2.385828723227042)--(-3.839514259733819,0.)); /* dots and labels */ dot((3.996177869179178,0.),blue); label("$B$", (4.040279615036859,0.10218054796102663), NE * labelscalefactor,blue); dot((-1.154148865691539,0.),blue); label("$C$", (-1.3803811057738653,-0.14328333373606214), NE * labelscalefactor,blue); dot((1.4210145017438194,1.5681827789938092),linewidth(4.pt)); label("$F$", (1.4629088572174203,1.6465574703052102), NE * labelscalefactor); dot((0.07833180472267817,2.385828723227042),linewidth(3.pt) + blue); label("$A$", (-0.04055741817725232,2.5568193649319144), NE * labelscalefactor,blue); dot((-3.839514259733819,0.),linewidth(3.pt)); label("$E$", (-4.049800819229713,-0.06146203983703255), NE * labelscalefactor); dot((1.4210145017438194,-2.40054783156011),linewidth(4.pt) + uuuuuu); label("$M$", (1.4117705485305265,-2.6490604593938434), NE * labelscalefactor,uuuuuu); dot((-0.6807342461448075,-3.3262298939043657),linewidth(4.pt)); label("$K$", (-0.7871767250058992,-3.5490946922831688), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/asy]