Let $\omega_1, \omega_2$ be two circles with different radii, and let $H$ be the exsimilicenter of the two circles. A point X outside both circles is given. The tangents from $X$ to $\omega_1$ touch $\omega_1$ at $P, Q$, and the tangents from $X$ to $\omega_2$ touch $\omega_2$ at $R, S$. If $PR$ passes through $H$ and is not a common tangent line of $\omega_1, \omega_2$, prove that $QS$ also passes through $H$.
Problem
Source: 2016 Thailand October Camp 1.5
Tags: geometry, collinear, circles
ineqcfe
23.10.2022 19:16
Bumppppppp this
demmy
27.01.2024 19:56
[asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(25cm); 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 = -27.08522313775581, xmax = 38.90404885552642, ymin = -17.197436711644208, ymax = 21.660357597988902; /* image dimensions */ pen qqccqq = rgb(0,0.8,0); pen qqqqcc = rgb(0,0,0.8); /* draw figures */ draw(circle((-3.2,0.14), 2.9600675667964067), linewidth(2) + qqccqq); draw(circle((7.401335383219947,0.28802233106575814), 5.001537660631567), linewidth(2) + qqccqq); draw((xmin, -0.18175968277464533*xmin-3.4501965124946192)--(xmax, -0.18175968277464533*xmax-3.4501965124946192), linewidth(2)); /* line */ draw((xmin, 0.2107603121414826*xmin + 3.839529236029733)--(xmax, 0.2107603121414826*xmax + 3.839529236029733), linewidth(2)); /* line */ draw((-18.571603594568455,-0.07462773452939447)--(10.845266533213193,3.9149655626068984), linewidth(2)); draw((-1.1617730403958257,2.2865392754717355)--(1.3383037366046626,-0.08738602195514038), linewidth(2)); draw((1.3383037366046626,-0.08738602195514038)--(3.115785269055124,2.866672983965016), linewidth(2)); draw((1.3383037366046626,-0.08738602195514038)--(-1.386541576677152,-2.1995231451899357), linewidth(2)); draw((1.3383037366046626,-0.08738602195514038)--(3.4666277082918397,-2.799606934823808), linewidth(2)); draw((xmin, -0.12364781741539978*xmin-2.3709659849017726)--(xmax, -0.12364781741539978*xmax-2.3709659849017726), linewidth(2)); /* line */ draw(circle((-0.15241169529121587,10.904256231163188), 8.676626985955046), linewidth(2) + qqqqcc); draw(circle((1.3383037366046626,-0.08738602195514038), 3.447594118898051), linewidth(2) + red); /* dots and labels */ dot((-3.7293480650007775,-2.77235139124384),linewidth(4pt) + dotstyle); dot((-3.8104539195443876,3.0364367785458204),linewidth(4pt) + dotstyle); dot((-0.9501306605137092,-1.783561268907106),linewidth(4pt) + dotstyle); dot((-1.004713142591101,2.1256272595046015),linewidth(4pt) + dotstyle); dot((-18.571603594568455,-0.07462773452939447),linewidth(4pt) + dotstyle); label("$H$", (-18.347967612512868,0.3920382273317282), NE * labelscalefactor); dot((-3.8104539195443876,3.0364367785458204),linewidth(4pt) + dotstyle); label("$A_{1}$", (-3.5751079415428912,3.4960632165627756), NE * labelscalefactor); dot((6.3698696481368104,5.182044951371602),linewidth(4pt) + dotstyle); label("$A_{2}$", (6.599196189825537,5.622895153628493), NE * labelscalefactor); dot((-3.7293480650007775,-2.77235139124384),linewidth(4pt) + dotstyle); label("$A_{3}$", (-3.5176259972978716,-2.309613152184183), NE * labelscalefactor); dot((6.506911785467683,-4.632890734463828),linewidth(4pt) + dotstyle); label("$A_{4}$", (6.714160078315575,-4.149035368024804), NE * labelscalefactor); dot((10.845266533213193,3.9149655626068984),dotstyle); label("$E$", (11.082787840937048,4.473256268728105), NE * labelscalefactor); dot((-1.1617730403958257,2.2865392754717355),linewidth(4pt) + dotstyle); label("$P$", (-0.9309385062720003,2.748797941377523), NE * labelscalefactor); dot((3.115785269055124,2.866672983965016),linewidth(4pt) + dotstyle); label("$R$", (3.322725367859433,3.323617383827717), NE * labelscalefactor); dot((1.3383037366046626,-0.08738602195514038),linewidth(4pt) + dotstyle); label("$X$", (1.5407850962638325,0.3920382273317282), NE * labelscalefactor); dot((-1.386541576677152,-2.1995231451899357),linewidth(4pt) + dotstyle); label("$Q$", (-1.1608662832520777,-1.7347937097339894), NE * labelscalefactor); dot((3.4666277082918397,-2.799606934823808),linewidth(4pt) + dotstyle); label("$S'$", (3.7250989775745684,-2.367095096429203), NE * labelscalefactor); dot((-5.736323578780784,1.6661266997600945),linewidth(4pt) + dotstyle); label("$F$", (-5.529494045873549,2.11649655468231), NE * labelscalefactor); dot((10.465481858111072,-3.664999974857669),linewidth(4pt) + dotstyle); label("$T'$", (10.680414231221912,-3.2293242601044936), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy] Let $HQ$ meets $\omega_2$ at $S'$ and $K'$ , $S'$ is closer to $X$. Then $$\angle{RS'Q} = \angle{RET} = \angle{HET} = \angle{HPQ}$$Thus $\square$ $PRS'Q$ is concyclic. Hence, $S' \equiv S$ $\implies$ $H \in QS$