Incenter-Related Configurations -Part (V)

by AlastorMoody, Nov 18, 2019, 2:40 AM

Here's the link to Part (IV). We'll dive deeper as we take down some Monsters in the Incenter Configuration! :diablo:

Thanks to Aryan-23 for notifying me this problem

USAJMO 2014 P6 wrote:

Let $ABC$ be a triangle with incenter $I$ , incircle $\gamma$ and circumcircle $\Gamma$ . Let $M,N,P$ be the midpoints of sides $\overline{BC}$ , $\overline{CA}$ , $\overline{AB}$ and let $E,F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$ , respectively. Let $U,V$ be the intersections of line $EF$ with line $MN$ and line $MP$ , respectively, and let $X$ be the midpoint of arc $BAC$ of $\Gamma$ .

(a) Prove that $I$ lies on ray $CV$ .

(b) Prove that line $XI$ bisects $\overline{UV}$ .

Part (a) trivially follows from Right-Angles on Incircle Chord Lemma. Also, $M$ is the center of $BVUC$ . By Reim's Theorem, there exists a Homothety at $I$ taking $UV$ to $M_BM_C$ , where $BI$ $\cap$ $\odot (ABC)$ $=$ $M_B$ and $CI$ $\cap$ $\odot (ABC)$ $=$ $M_C$ . Let $IX$ $\cap$ $M_BM_C$ $=$ $J$ , then, since, $I$ is orthocenter WRT $\Delta M_AM_BM_C$ and since, $XM_CIM_B$ is parallelogram, it follows $J$ is midpoint of $M_BM_C$ . Hence, $IX$ $\cap$ $EF$ must be the midpoint of $UV$ which ofcourse lies on perpendicular from $M$ to $EF$ $\qquad \square$
Aryan-23's Approach Using Lengths

This above problem, especially part (b), will heavily reduce our burden in below problems!

AlastorMoody wrote:

Let $AT$ be the isogonal conjugate of the $A-$ Nagel cevian, where $T \in \odot (ABC)$ . Let $(I)$ be the incircle. Let $(I)$ touch $BC,$ $CA,$ and $AB$ at $D,$ $E,$ $F$ . Let $D'$ be the first intersection of ray $TA$ with $(I)$ and Let $AT$ intersect $EF$ at $L$ . Let $EF$ intersect $ID$ at $J$ , prove, $JLD'D$ cyclic

STEMS 2019 Cat B P6 wrote:

In triangle $ABC$ , with circumcircle $\Gamma$ , the incircle $\omega$ has center $I$ and touches sides $BC, CA, AB$ at points $D, E, F$ respectively. Point $Q$ lies on $EF$ and point $R$ lies on $\omega$ , such that and $DQ \perp EF$ and $D, Q, R$ are collinear. Ray $AR$ meets $\omega$ again at $P$ and $\Gamma$ again at $S$ . Ray $AQ$ meets $BC$ at $T$ . Let $M$ be the midpoint of $BC$ and $O$ be the circumcenter of triangle $MPD$ . Prove that $O, T, I, S$ are collinear.

Mathematical Reflections O451 wrote:

Let $ABC$ be a triangle, $\Gamma$ its circumcircle, $\omega$ its incircle and $I$ the incenter. Let $M$ be the midpoint of $BC$ . The incircle $\omega$ is tangent to $AB$ and $AC$ at $F$ and $E$ ; respectively. Suppose $EF$ meets $\Gamma$ at distinct points $P$ and $Q$ . Let $J$ denote the point on $EF$ such that $MJ$ is perpendicular on $EF$ . Show that $IJ$ and the radical axis of $(MPQ)$ and $(AJI)$ intersect on $\Gamma$ .

Taiwan TST R2 2019 Day 1 P2 wrote:

Let $ABC$ be a scalene triangle and $I$ be its incenter, $\Omega$ be its circumcircle. Let $M$ be the midpoint of $BC$ . The incircle $\omega$ touches $CA,AB$ at $E,F$ , respectively. Suppose the line $EF$ intersects $\Omega$ at two points $P,Q$ , and $R$ be the point on the circumcircle $\Gamma$ of triangle $MPQ$ such that $MR$ is perpendicular to $PQ$ . Prove that $AR,\Gamma,\omega$ intersects at a point.

Let's try the first result, which is indeed a crucial step in proving the Taiwan TST problem.
$[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(14cm); 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 = -9.8030712381575, xmax = 12.410272150317091, ymin = -9.35947015194124, ymax = 3.258437199052897; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen ubqqys = rgb(0.29411764705882354,0,0.5098039215686274); pen qqqqcc = rgb(0,0,0.8); pen wwccqq = rgb(0.4,0.8,0); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); draw((-3.3647375827835124,1.7629912237603584)--(-5.96,-4.53)--(5.1,-4.73)--cycle, linewidth(0.4) + rvwvcq); /* draw figures */ draw((-3.3647375827835124,1.7629912237603584)--(-5.96,-4.53), linewidth(0.4) + rvwvcq); draw((-5.96,-4.53)--(5.1,-4.73), linewidth(0.4) + rvwvcq); draw((5.1,-4.73)--(-3.3647375827835124,1.7629912237603584), linewidth(0.4) + rvwvcq); draw(circle((-0.4030572637793516,-3.1400666869981144), 5.728134704092408), linewidth(0.4)); draw((-0.2994912894680226,2.587131692418541)--(-0.5066232380906821,-8.86726506641477), linewidth(0.4)); draw((-0.2994912894680226,2.587131692418541)--(-4.466063214608107,-7.177827184075416), linewidth(0.4)); draw((-3.3647375827835124,1.7629912237603584)--(-0.5066232380906821,-8.86726506641477), linewidth(0.4)); draw((-3.3647375827835124,1.7629912237603584)--(-4.466063214608107,-7.177827184075416), linewidth(0.4) + ubqqys); draw(circle((-2.3157953254666253,-2.138365994357423), 2.4571311144143166), linewidth(0.4) + qqqqcc); draw((-4.587338065571309,-1.2015699155330901)--(-0.8203120282457429,-0.18874474366376615), linewidth(0.4) + wwccqq); draw((-0.8203120282457429,-0.18874474366376615)--(-2.3602208131726323,-4.595095464499592), linewidth(0.4) + wwccqq); draw((-2.3602208131726323,-4.595095464499592)--(-4.587338065571309,-1.2015699155330901), linewidth(0.4) + wwccqq); draw((-2.287674213911024,-0.5832685253326907)--(-2.3602208131726323,-4.595095464499592), linewidth(0.4) + wwccqq); draw(circle((-2.506741820946712,-2.5858764920697612), 2.014554363846041), linewidth(0.4) + linetype("4 4") + dtsfsf); /* dots and labels */ dot((-3.3647375827835124,1.7629912237603584),dotstyle); label("$A$", (-3.284729489736267,1.9474857859625971), NE * labelscalefactor); dot((-5.96,-4.53),dotstyle); label("$B$", (-5.888424657401731,-4.3523640602769), NE * labelscalefactor); dot((5.1,-4.73),dotstyle); label("$C$", (5.1818317198052775,-4.5526483039434735), NE * labelscalefactor); dot((-2.3157953254666253,-2.138365994357423),dotstyle); label("$I$", (-2.2468929543731098,-1.9489531362780166), NE * labelscalefactor); dot((-0.5066232380906821,-8.86726506641477),dotstyle); label("$M_A$", (-0.4261271028587992,-8.685786786880946), NE * labelscalefactor); dot((-0.2994912894680226,2.587131692418541),dotstyle); label("$M_{BC}$", (-0.225842859192225,2.7668304191440347), NE * labelscalefactor); dot((-4.466063214608107,-7.177827184075416),dotstyle); label("$T$", (-4.395396659159997,-6.992474544972643), NE * labelscalefactor); dot((-2.3602208131726323,-4.595095464499592),dotstyle); label("$D$", (-2.283308271403396,-4.406987035822329), NE * labelscalefactor); dot((-0.8203120282457429,-0.18874474366376615),dotstyle); label("$E$", (-0.753864956131375,-0.0007336751577098196), NE * labelscalefactor); dot((-4.587338065571309,-1.2015699155330901),dotstyle); label("$F$", (-4.522850268765998,-1.020362552005721), NE * labelscalefactor); dot((-4.058655585491474,-3.87039700662281),dotstyle); label("$D'$", (-3.994828171826848,-3.69688835373175), NE * labelscalefactor); dot((-3.700541138956399,-0.9631403849929725),dotstyle); label("$L$", (-3.630675001523986,-0.7836629913088612), NE * labelscalefactor); dot((-2.287674213911024,-0.5832685253326907),dotstyle); label("$J$", (-2.2104776373428234,-0.40130216249085704), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]$
Recall, we already earlier proved that, $D'L \cap (I)$ is reflection of orthocenter WRT $\Delta DEF$ over $EF$ and $DI$ $\cap$ $(I)$ is the $D-$ antipode. Hence, by Converse of Reim's theorem, we get $DD'LJ$ is cyclic $\qquad \square$

Let's take a look at the STEMS problem! This problem is tough but beautiful tough! Luckily, since we have already done the basics on the configuration earlier, we won't have any much difficulties,
$[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(18cm); 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 = -11.515515234203619, xmax = 13.331794051704854, ymin = -8.868184822118293, ymax = 5.2459015181560025; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen qqqqcc = rgb(0,0,0.8); pen ttffqq = rgb(0.2,1,0); pen ubqqys = rgb(0.29411764705882354,0,0.5098039215686274); pen ffqqff = rgb(1,0,1); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); draw((-4.620593695741067,2.849901959146017)--(-5.96,-4.53)--(5.1,-4.73)--cycle, linewidth(0.4) + rvwvcq); draw((-4.562293637057545,-3.9115788828959563)--(-2.8426691930227572,-4.58637126233232)--(-0.43,-4.63)--cycle, linewidth(0.4) + ffqqff); /* draw figures */ draw((-4.620593695741067,2.849901959146017)--(-5.96,-4.53), linewidth(0.4) + rvwvcq); draw((-5.96,-4.53)--(5.1,-4.73), linewidth(0.4) + rvwvcq); draw((5.1,-4.73)--(-4.620593695741067,2.849901959146017), linewidth(0.4) + rvwvcq); draw(circle((-0.3775890583776725,-1.731674928285277), 6.244512401151261), linewidth(0.4)); draw((-0.2646868582142609,4.511816740751381)--(-0.490491258541085,-7.975166597321936), linewidth(0.4)); draw((-4.620593695741067,2.849901959146017)--(-0.490491258541085,-7.975166597321936), linewidth(0.4)); draw(circle((-2.7947366275701775,-1.9357003928046626), 2.651104220771672), linewidth(0.4) + qqqqcc); draw((-2.8426691930227572,-4.58637126233232)--(-5.403227101238288,-1.4622755960564502), linewidth(0.4) + ttffqq); draw((-5.403227101238288,-1.4622755960564502)--(-1.164512226750342,0.15492678399300508), linewidth(0.4) + ttffqq); draw((-1.164512226750342,0.15492678399300508)--(-2.8426691930227572,-4.58637126233232), linewidth(0.4) + ttffqq); draw((-4.54096052412599,-6.385734786766322)--(-0.2646868582142609,4.511816740751381), linewidth(0.4)); draw((-4.620593695741067,2.849901959146017)--(-4.54096052412599,-6.385734786766322), linewidth(0.4)); draw((-4.596101855188915,0.009405492775725544)--(-2.8426691930227572,-4.58637126233232), linewidth(0.4)); draw((-3.8278869149203847,-4.568555390326938)--(-4.562293637057545,-3.9115788828959563), linewidth(0.4) + rvwvcq); draw((-4.620593695741067,2.849901959146017)--(-3.8278869149203847,-4.568555390326938), linewidth(0.4) + ubqqys); draw((-4.562293637057545,-3.9115788828959563)--(-2.8426691930227572,-4.58637126233232), linewidth(0.4) + ffqqff); draw((-2.8426691930227572,-4.58637126233232)--(-0.43,-4.63), linewidth(0.4) + ffqqff); draw((-0.43,-4.63)--(-4.562293637057545,-3.9115788828959563), linewidth(0.4) + ffqqff); draw((-3.5567082582136305,-7.106350382800299)--(-0.2646868582142609,4.511816740751381), linewidth(0.4) + dtsfsf); draw((-3.8278869149203847,-4.568555390326938)--(-3.5567082582136305,-7.106350382800299), linewidth(0.4) + ubqqys); draw(circle((-3.7076651616556227,0.4571007831706772), 2.561041970779934), linewidth(0.4)); draw((-4.54096052412599,-6.385734786766322)--(-2.8426691930227572,-4.58637126233232), linewidth(0.4)); draw(circle((-3.357415583887273,-5.8016687306679335), 1.3198150565329196), linewidth(0.4)); /* dots and labels */ dot((-4.620593695741067,2.849901959146017),dotstyle); label("$A$", (-4.529755328411319,3.046303646944424), NE * labelscalefactor); dot((-5.96,-4.53),dotstyle); label("$B$", (-5.873954027485056,-4.326422551005496), NE * labelscalefactor); dot((5.1,-4.73),dotstyle); label("$C$", (5.1851352694397805,-4.530089020562124), NE * labelscalefactor); dot((-2.7947366275701775,-1.9357003928046626),dotstyle); label("$I$", (-2.7171237493573406,-1.7398583876363254), NE * labelscalefactor); dot((-0.490491258541085,-7.975166597321936),dotstyle); label("$M_A$", (-0.41569264336745726,-7.768385886512503), NE * labelscalefactor); dot((-0.2646868582142609,4.511816740751381),dotstyle); label("$M_{BC}$", (-0.19165952685516777,4.716368697308771), NE * labelscalefactor); dot((-4.54096052412599,-6.385734786766322),dotstyle); label("$S$", (-4.468655387544331,-6.179787423970808), NE * labelscalefactor); dot((-2.8426691930227572,-4.58637126233232),dotstyle); label("$D$", (-2.757857043268666,-4.387522491872485), NE * labelscalefactor); dot((-1.164512226750342,0.15492678399300508),dotstyle); label("$E$", (-1.0877919929043258,0.3579062487969392), NE * labelscalefactor); dot((-5.403227101238288,-1.4622755960564502),dotstyle); label("$F$", (-5.324054559682164,-1.2510588607004192), NE * labelscalefactor); dot((-4.208518560529974,-1.0064568980053574),dotstyle); label("$Q$", (-4.122422389298066,-0.8029926276758382), NE * labelscalefactor); dot((-4.596101855188915,0.009405492775725544),dotstyle); label("$R$", (-4.509388681455657,0.21533972010729985), NE * labelscalefactor); dot((-4.562293637057545,-3.9115788828959563),dotstyle); label("$P$", (-4.489022034499994,-3.7154231423356134), NE * labelscalefactor); dot((-3.8278869149203847,-4.568555390326938),dotstyle); label("$T$", (-3.7558227440961374,-4.367155844916822), NE * labelscalefactor); dot((-0.43,-4.63),dotstyle); label("$M$", (-0.3545927025004692,-4.4282557857838105), NE * labelscalefactor); dot((-3.5567082582136305,-7.106350382800299),dotstyle); label("$V$", (-3.47068968671686,-6.892620067419005), NE * labelscalefactor); dot((-6.266299237125552,0.3460708595441685),dotstyle); label("$L$", (-6.179453731819996,0.5412060713979041), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]$
Highly refer to Part (I) & (II). We already know, $S,I, M_{BC}$ are collinear. Let $AT$ $\cap$ $\odot (ABC)$ $=$ $V$ . We want to show firstly, $AV$ & $SM_{BC}$ concur on $BC$ . But this is obvious, from Pascal on $AVM_ASM_{BC}L$ , where $L$ $= \odot (AEF)$ $\cap$ $\odot (ABC)$ . Then, from Shooting Lemma, $STDV$ is cyclic. Also, $\angle TSD$ $=$ $\angle TVD$ $=$ $\angle TSP$ $\implies$ Line $SD$ is reflection of Line $AS$ over $SM_{BC}$ . Since, $IP=ID$ and also from Reflection argument, $TP$ is tangent to $(I)$ $\implies$ $IT$ is perpendicular bisector of $PD$ $\implies$ $SP=SD$ . Moreover, since, $O$ the circumcenter WRT $\Delta MPD$ must lie on perpendicular bisector of $PD$ $\implies$ $O,T,I,S$ are collinear $\qquad \square$

Let's take a look at the Mathematical Reflections Problem which was proposed by AoPS User: tworigami Apparently, this problem is beautiful, tough and cleverly stated. Anyway,
$[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(19cm); 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 = -15.017491981658836, xmax = 11.131477483886478, ymin = -8.250811492972314, ymax = 6.602660506718562; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen qqqqcc = rgb(0,0,0.8); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); draw((-4.618108299402482,3.6910718535547)--(-5.96,-4.53)--(5.1,-4.73)--cycle, linewidth(0.4) + rvwvcq); /* draw figures */ draw((-4.618108299402482,3.6910718535547)--(-5.96,-4.53), linewidth(0.4) + rvwvcq); draw((-5.96,-4.53)--(5.1,-4.73), linewidth(0.4) + rvwvcq); draw((5.1,-4.73)--(-4.618108299402482,3.6910718535547), linewidth(0.4) + rvwvcq); draw(circle((-0.3683841730197502,-1.2226447679921784), 6.496519544127112), linewidth(0.4)); draw((-0.2509256265684152,5.272812850766571)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw((-4.618108299402482,3.6910718535547)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw(circle((-2.6431144226071637,-1.7618809705153757), 2.827636594269825), linewidth(0.4) + qqqqcc); draw((-6.837851891540301,-1.8148893342365993)--(4.979844423017278,2.465338023691826), linewidth(0.4)); draw(circle((-1.44702002458779,1.7554659401255956), 6.465949659788316), linewidth(0.4)); draw((-2.0744613450109326,-0.08964661424579373)--(-0.43,-4.63), linewidth(0.4)); draw(circle((-5.7066256861518205,0.21268804151315282), 3.6447255102583913), linewidth(0.4)); draw((-4.235015582748999,-6.4431742170539055)--(-0.2509256265684152,5.272812850766571), linewidth(0.4)); draw((-4.235015582748999,-6.4431742170539055)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw((-13.93604048454686,-4.385767803172747)--(-5.96,-4.53), linewidth(0.4) + rvwvcq); draw((-13.93604048454686,-4.385767803172747)--(-6.837851891540301,-1.8148893342365993), linewidth(0.4)); draw((-3.2969221131279314,-7.021647416773472)--(-0.2509256265684152,5.272812850766571), linewidth(0.4)); draw((-13.93604048454686,-4.385767803172747)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw((-13.93604048454686,-4.385767803172747)--(-0.2509256265684152,5.272812850766571), linewidth(0.4)); draw((-7.740992957065166,3.2368199586366244)--(-4.235015582748999,-6.4431742170539055), linewidth(0.4) + linetype("4 4") + dtsfsf); /* dots and labels */ dot((-4.618108299402482,3.6910718535547),dotstyle); label("$A$", (-4.536470613911576,3.902029234047494), NE * labelscalefactor); dot((-5.96,-4.53),dotstyle); label("$B$", (-5.865352668717978,-4.307032491611389), NE * labelscalefactor); dot((5.1,-4.73),dotstyle); label("$C$", (5.194375400315942,-4.521368306902744), NE * labelscalefactor); dot((-2.6431144226071637,-1.7618809705153757),dotstyle); label("$I$", (-2.5645811132311103,-1.542100474352914), NE * labelscalefactor); dot((-0.48584271947108304,-7.718102386750928),dotstyle); label("$M_A$", (-0.39978937878842435,-7.500636139452573), NE * labelscalefactor); dot((-0.2509256265684152,5.272812850766571),dotstyle); label("$M_{BC}$", (-0.16401998196793383,5.488114267203518), NE * labelscalefactor); dot((-4.235015582748999,-6.4431742170539055),dotstyle); label("$T$", (-4.150666146387137,-6.236054829233581), NE * labelscalefactor); dot((-0.43,-4.63),dotstyle); label("$M$", (-0.3354886342010179,-4.414200399257067), NE * labelscalefactor); dot((-2.694238733401516,-4.589055357443012),dotstyle); label("$D$", (-2.6074482762893814,-4.3713332361987955), NE * labelscalefactor); dot((-0.791370694583703,0.37507337503804045),dotstyle); label("$E$", (-0.6998595201963214,0.5798240970314968), NE * labelscalefactor); dot((-5.433819446773386,-1.3063656827946843),dotstyle); label("$F$", (-5.350946712018725,-1.0919952622410694), NE * labelscalefactor); dot((-6.837851891540301,-1.8148893342365993),dotstyle); label("$P$", (-6.744129511412533,-1.6064012189403205), NE * labelscalefactor); dot((4.979844423017278,2.465338023691826),dotstyle); label("$Q$", (5.065773911141129,2.6803150868867722), NE * labelscalefactor); dot((-2.0744613450109326,-0.08964661424579373),dotstyle); label("$J$", (-1.9858744119444516,0.12971888491965208), NE * labelscalefactor); dot((-5.3325830164503545,-3.412793461180097),dotstyle); label("$X$", (-5.243778804373048,-3.192486252096345), NE * labelscalefactor); dot((-7.740992957065166,3.2368199586366244),dotstyle); label("$Y$", (-7.66577351716536,3.4519240219356493), NE * labelscalefactor); dot((-13.93604048454686,-4.385767803172747),dotstyle); label("$G$", (-13.86007857908552,-4.178431002436576), NE * labelscalefactor); dot((-3.2969221131279314,-7.021647416773472),dotstyle); label("$V$", (-3.2075885591051754,-6.814761530520238), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]$
Let $G$ be orthocenter WRT $\Delta M_ADM_{BC}$ , $GP \times GQ=GV \times GM_A = GD \times GM$ Hence, $PDMQ$ is cyclic. Let $AT$ $\cap$ $\odot (I)$ $=$ $D'$ . So from the above STEMS example, we know $TD=TD'$ . Let $\Delta M_AM_BM_C$ be circum-Incentral Triangle WRT $\Delta ABC$ . Since, $I$ is Orthocenter WRT $\Delta M_AM_BM_C$ $\implies$ $IM_CM_{BC}M_B$ is parallelogram & $PM_CM_BQ$ is Isosceles Trapezium. Suppose if $Z$ is midpoint of $M_BM_C$ $\implies$ $ZD=ZD'$ . Hence, $Z$ is the center of $\odot (PD'DMQ)$ . Let $DI \cap EF=I'$ .
$[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(18cm); 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 = -14.525882318406245, xmax = 9.402893439360872, ymin = -7.792735443295693, ymax = 5.689086995836438; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen qqqqcc = rgb(0,0,0.8); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); pen ttffqq = rgb(0.2,1,0); draw((-4.618108299402482,3.6910718535547)--(-5.96,-4.53)--(5.1,-4.73)--cycle, linewidth(0.4) + rvwvcq); draw((-6.780053115248595,-0.1760944526782966)--(-2.6431144226071637,-1.7618809705153757)--(3.8860130660730166,3.6870263329294897)--(-0.2509256265684152,5.272812850766571)--cycle, linewidth(0.4) + ttffqq); /* draw figures */ draw((-4.618108299402482,3.6910718535547)--(-5.96,-4.53), linewidth(0.4) + rvwvcq); draw((-5.96,-4.53)--(5.1,-4.73), linewidth(0.4) + rvwvcq); draw((5.1,-4.73)--(-4.618108299402482,3.6910718535547), linewidth(0.4) + rvwvcq); draw(circle((-0.3683841730197502,-1.2226447679921784), 6.496519544127112), linewidth(0.4)); draw((-0.2509256265684152,5.272812850766571)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw((-4.618108299402482,3.6910718535547)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw(circle((-2.6431144226071637,-1.7618809705153757), 2.827636594269825), linewidth(0.4) + qqqqcc); draw((-6.837851891540301,-1.8148893342365993)--(4.979844423017278,2.465338023691826), linewidth(0.4)); draw(circle((-1.44702002458779,1.7554659401255956), 6.465949659788316), linewidth(0.4)); draw((-2.0744613450109326,-0.08964661424579373)--(-0.43,-4.63), linewidth(0.4)); draw(circle((-5.7066256861518205,0.21268804151315282), 3.6447255102583913), linewidth(0.4)); draw((-4.235015582748999,-6.4431742170539055)--(-0.2509256265684152,5.272812850766571), linewidth(0.4)); draw((-4.235015582748999,-6.4431742170539055)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw((-13.93604048454686,-4.385767803172747)--(-5.96,-4.53), linewidth(0.4) + rvwvcq); draw((-13.93604048454686,-4.385767803172747)--(-6.837851891540301,-1.8148893342365993), linewidth(0.4)); draw((-3.2969221131279314,-7.021647416773472)--(-0.2509256265684152,5.272812850766571), linewidth(0.4)); draw((-13.93604048454686,-4.385767803172747)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw((-13.93604048454686,-4.385767803172747)--(-0.2509256265684152,5.272812850766571), linewidth(0.4)); draw((-7.740992957065166,3.2368199586366244)--(-4.235015582748999,-6.4431742170539055), linewidth(0.4) + linetype("4 4") + dtsfsf); draw((-4.618108299402482,3.6910718535547)--(-4.235015582748999,-6.4431742170539055), linewidth(0.4)); draw((-4.618108299402482,3.6910718535547)--(-3.2969221131279314,-7.021647416773472), linewidth(0.4)); draw((-6.780053115248595,-0.1760944526782966)--(3.8860130660730166,3.6870263329294897), linewidth(0.4) + dtsfsf); draw((-6.780053115248595,-0.1760944526782966)--(-2.6431144226071637,-1.7618809705153757), linewidth(0.4) + ttffqq); draw((-2.6431144226071637,-1.7618809705153757)--(3.8860130660730166,3.6870263329294897), linewidth(0.4) + ttffqq); draw((3.8860130660730166,3.6870263329294897)--(-0.2509256265684152,5.272812850766571), linewidth(0.4) + ttffqq); draw((-0.2509256265684152,5.272812850766571)--(-6.780053115248595,-0.1760944526782966), linewidth(0.4) + ttffqq); draw((-4.235015582748999,-6.4431742170539055)--(-2.694238733401516,-4.589055357443012), linewidth(0.4)); draw(circle((-1.5232123491588767,-2.4579696086554956), 2.431630987892209), linewidth(0.4) + linetype("4 4")); draw((-2.616424698317758,-0.2859392173109916)--(-2.694238733401516,-4.589055357443012), linewidth(0.4)); draw(circle((-2.8715322243189374,-2.4335876940956407), 2.1627468251410904), linewidth(0.4) + linetype("4 4")); /* dots and labels */ dot((-4.618108299402482,3.6910718535547),dotstyle); label("$A$", (-4.545831941386301,3.879838097078445), NE * labelscalefactor); dot((-5.96,-4.53),dotstyle); label("$B$", (-5.888177898529334,-4.329871959651366), NE * labelscalefactor); dot((5.1,-4.73),dotstyle); label("$C$", (5.181312675592202,-4.543869141224892), NE * labelscalefactor); dot((-2.6431144226071637,-1.7618809705153757),dotstyle); label("$I$", (-2.5614944395226864,-1.5673628884294863), NE * labelscalefactor); dot((-0.48584271947108304,-7.718102386750928),dotstyle); label("$M_A$", (-0.40206833455345875,-7.520375394020297), NE * labelscalefactor); dot((-0.2509256265684152,5.272812850766571),dotstyle); label("$M_{BC}$", (-0.16861686374597465,5.241638343455429), NE * labelscalefactor); dot((-4.235015582748999,-6.4431742170539055),dotstyle); label("$T$", (-4.156746156707161,-6.255846593813099), NE * labelscalefactor); dot((-0.43,-4.63),dotstyle); label("$M$", (-0.3437054668515877,-4.42714340582115), NE * labelscalefactor); dot((-2.694238733401516,-4.589055357443012),dotstyle); label("$D$", (-2.6198573072245575,-4.388234827353236), NE * labelscalefactor); dot((-0.791370694583703,0.37507337503804045),dotstyle); label("$E$", (-0.7133369622967708,0.5726089273057726), NE * labelscalefactor); dot((-5.433819446773386,-1.3063656827946843),dotstyle); label("$F$", (-5.362912089212496,-1.1199142360484775), NE * labelscalefactor); dot((-6.837851891540301,-1.8148893342365993),dotstyle); label("$P$", (-6.7636209140574,-1.625725756131357), NE * labelscalefactor); dot((4.979844423017278,2.465338023691826),dotstyle); label("$Q$", (5.06458694018846,2.6542178753391608), NE * labelscalefactor); dot((-2.0744613450109326,-0.08964661424579373),dotstyle); label("$J$", (-1.9973200517379333,0.10570598569080704), NE * labelscalefactor); dot((-5.3325830164503545,-3.412793461180097),dotstyle); label("$X$", (-5.246186353808754,-3.2209774733158225), NE * labelscalefactor); dot((-7.740992957065166,3.2368199586366244),dotstyle); label("$Y$", (-7.658518218819422,3.432389444697437), NE * labelscalefactor); dot((-13.93604048454686,-4.385767803172747),dotstyle); label("$G$", (-13.864436484451707,-4.193691935013668), NE * labelscalefactor); dot((-3.2969221131279314,-7.021647416773472),dotstyle); label("$V$", (-3.222940273477225,-6.820020981597849), NE * labelscalefactor); dot((-4.326081493888807,-4.034137678335941),dotstyle); label("$D'$", (-4.254017602876946,-3.843514728802443), NE * labelscalefactor); dot((-6.780053115248595,-0.1760944526782966),dotstyle); label("$M_C$", (-6.705258046355529,0.027888828754979448), NE * labelscalefactor); dot((3.8860130660730166,3.6870263329294897),dotstyle); label("$M_B$", (3.955692453852911,3.879838097078445), NE * labelscalefactor); dot((-1.4470200245877902,1.7554659401255965),dotstyle); label("$Z$", (-1.374782796251309,1.9538634629167124), NE * labelscalefactor); dot((-2.616424698317758,-0.2859392173109916),dotstyle); label("$I'$", (-2.5420401502887295,-0.08883690664876194), NE * labelscalefactor); dot((-4.442765056660414,-0.9474177118156701),dotstyle); label("$L$", (-4.370743338280688,-0.7502827406032965), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]$
Our next step is to show, $J$ which is the foot from $M$ to $EF$ lies on $TI$ . Redefine: $TI$ $\cap$ $EF$ $=$ $J$ . Taking Homothety at $I$ which sends $J$ to $Z$ and using Right Angle on Incircle Chords Lemma, we have $MJ$ $\perp$ $EF$ . Let $D'A$ $\cap$ $\odot (PQMD)$ $=$ $N$
$\angle JMD=\angle LI'D=180^{\circ}-\angle LD'D=\angle NMD$ $[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(19cm); 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 = -16.703984590817605, xmax = 14.263615087265507, ymin = -8.740605115142586, ymax = 8.706993727874869; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen qqqqcc = rgb(0,0,0.8); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); pen ttffqq = rgb(0.2,1,0); draw((-4.618108299402482,3.6910718535547)--(-5.96,-4.53)--(5.1,-4.73)--cycle, linewidth(0.4) + rvwvcq); draw((-6.780053115248595,-0.1760944526782966)--(-2.6431144226071637,-1.7618809705153757)--(3.8860130660730166,3.6870263329294897)--(-0.2509256265684152,5.272812850766571)--cycle, linewidth(0.4) + ttffqq); /* draw figures */ draw((-4.618108299402482,3.6910718535547)--(-5.96,-4.53), linewidth(0.4) + rvwvcq); draw((-5.96,-4.53)--(5.1,-4.73), linewidth(0.4) + rvwvcq); draw((5.1,-4.73)--(-4.618108299402482,3.6910718535547), linewidth(0.4) + rvwvcq); draw(circle((-0.3683841730197502,-1.2226447679921784), 6.496519544127112), linewidth(0.4)); draw((-0.2509256265684152,5.272812850766571)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw((-4.618108299402482,3.6910718535547)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw(circle((-2.6431144226071637,-1.7618809705153757), 2.827636594269825), linewidth(0.4) + qqqqcc); draw((-6.837851891540301,-1.8148893342365993)--(4.979844423017278,2.465338023691826), linewidth(0.4)); draw(circle((-1.44702002458779,1.7554659401255956), 6.465949659788316), linewidth(0.4)); draw((-2.0744613450109326,-0.08964661424579373)--(-0.43,-4.63), linewidth(0.4)); draw(circle((-5.7066256861518205,0.21268804151315282), 3.6447255102583913), linewidth(0.4)); draw((-4.235015582748999,-6.4431742170539055)--(-0.2509256265684152,5.272812850766571), linewidth(0.4)); draw((-4.235015582748999,-6.4431742170539055)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw((-13.93604048454686,-4.385767803172747)--(-5.96,-4.53), linewidth(0.4) + rvwvcq); draw((-13.93604048454686,-4.385767803172747)--(-6.837851891540301,-1.8148893342365993), linewidth(0.4)); draw((-3.2969221131279314,-7.021647416773472)--(-0.2509256265684152,5.272812850766571), linewidth(0.4)); draw((-13.93604048454686,-4.385767803172747)--(-0.48584271947108304,-7.718102386750928), linewidth(0.4)); draw((-13.93604048454686,-4.385767803172747)--(-0.2509256265684152,5.272812850766571), linewidth(0.4)); draw((-7.740992957065166,3.2368199586366244)--(-4.235015582748999,-6.4431742170539055), linewidth(0.4) + linetype("4 4") + dtsfsf); draw((-4.618108299402482,3.6910718535547)--(-4.235015582748999,-6.4431742170539055), linewidth(0.4)); draw((-4.618108299402482,3.6910718535547)--(-3.2969221131279314,-7.021647416773472), linewidth(0.4)); draw((-6.780053115248595,-0.1760944526782966)--(3.8860130660730166,3.6870263329294897), linewidth(0.4) + dtsfsf); draw((-6.780053115248595,-0.1760944526782966)--(-2.6431144226071637,-1.7618809705153757), linewidth(0.4) + ttffqq); draw((-2.6431144226071637,-1.7618809705153757)--(3.8860130660730166,3.6870263329294897), linewidth(0.4) + ttffqq); draw((3.8860130660730166,3.6870263329294897)--(-0.2509256265684152,5.272812850766571), linewidth(0.4) + ttffqq); draw((-0.2509256265684152,5.272812850766571)--(-6.780053115248595,-0.1760944526782966), linewidth(0.4) + ttffqq); draw((-4.235015582748999,-6.4431742170539055)--(-2.694238733401516,-4.589055357443012), linewidth(0.4)); draw(circle((-1.5232123491588767,-2.4579696086554956), 2.431630987892209), linewidth(0.4) + linetype("4 4")); draw((-2.616424698317758,-0.2859392173109916)--(-2.694238733401516,-4.589055357443012), linewidth(0.4)); draw(circle((-2.8715322243189374,-2.4335876940956407), 2.1627468251410904), linewidth(0.4) + linetype("4 4")); draw((-4.754955270674418,7.3111899623043834)--(-4.618108299402482,3.6910718535547), linewidth(0.4)); draw((-4.754955270674418,7.3111899623043834)--(-2.0744613450109326,-0.08964661424579373), linewidth(0.4)); /* dots and labels */ dot((-4.618108299402482,3.6910718535547),dotstyle); label("$A$", (-4.51836000204344,3.9485576797792), NE * labelscalefactor); dot((-5.96,-4.53),dotstyle); label("$B$", (-5.852736248334826,-4.284291990735529), NE * labelscalefactor); dot((5.1,-4.73),dotstyle); label("$C$", (5.199927376607032,-4.485707273194605), NE * labelscalefactor); dot((-2.6431144226071637,-1.7618809705153757),dotstyle); label("$I$", (-2.5545609980674375,-1.5148318569232349), NE * labelscalefactor); dot((-0.48584271947108304,-7.718102386750928),dotstyle); label("$M_A$", (-0.38934671163235834,-7.456582689465975), NE * labelscalefactor); dot((-0.2509256265684152,5.272812850766571),dotstyle); label("$M_{BC}$", (-0.16275451886589656,5.534703029144423), NE * labelscalefactor); dot((-4.235015582748999,-6.4431742170539055),dotstyle); label("$T$", (-4.14070634743267,-6.19773717409675), NE * labelscalefactor); dot((-0.43,-4.63),dotstyle); label("$M$", (-0.33899289101758906,-4.3849996319650675), NE * labelscalefactor); dot((-2.694238733401516,-4.589055357443012),dotstyle); label("$D$", (-2.604914818682207,-4.334645811350298), NE * labelscalefactor); dot((-0.791370694583703,0.37507337503804045),dotstyle); label("$E$", (-0.691469635320974,0.625205519204447), NE * labelscalefactor); dot((-5.433819446773386,-1.3063656827946843),dotstyle); label("$F$", (-5.324021131879748,-1.0616474713903141), NE * labelscalefactor); dot((-6.837851891540301,-1.8148893342365993),dotstyle); label("$P$", (-6.733928109093288,-1.565185677538004), NE * labelscalefactor); dot((4.979844423017278,2.465338023691826),dotstyle); label("$Q$", (5.074042825070109,2.7148890747173597), NE * labelscalefactor); dot((-2.0744613450109326,-0.08964661424579373),dotstyle); label("$J$", (-1.9754920609975908,0.1720211336715261), NE * labelscalefactor); dot((-5.3325830164503545,-3.412793461180097),dotstyle); label("$X$", (-5.2233134906502094,-3.151331026903227), NE * labelscalefactor); dot((-7.740992957065166,3.2368199586366244),dotstyle); label("$Y$", (-7.640296880159135,3.495373294246279), NE * labelscalefactor); dot((-13.93604048454686,-4.385767803172747),dotstyle); label("$G$", (-13.833816815775757,-4.1332305288912226), NE * labelscalefactor); dot((-3.2969221131279314,-7.021647416773472),dotstyle); label("$V$", (-3.183983755752054,-6.776806111166594), NE * labelscalefactor); dot((-4.326081493888807,-4.034137678335941),dotstyle); label("$D'$", (-4.216237078354824,-3.7807537845878394), NE * labelscalefactor); dot((-6.780053115248595,-0.1760944526782966),dotstyle); label("$M_C$", (-6.6835742884785185,0.07131349244198812), NE * labelscalefactor); dot((3.8860130660730166,3.6870263329294897),dotstyle); label("$M_B$", (3.9914356818525696,3.9485576797792), NE * labelscalefactor); dot((-1.4470200245877902,1.7554659401255965),dotstyle); label("$Z$", (-1.3460693033129747,2.0099355861105943), NE * labelscalefactor); dot((-2.616424698317758,-0.2859392173109916),dotstyle); label("$I'$", (-2.504207177452668,-0.029394148787549854), NE * labelscalefactor); dot((-4.442765056660414,-0.9474177118156701),dotstyle); label("$L$", (-4.342121629891747,-0.6839938167795466), NE * labelscalefactor); dot((-4.754955270674418,7.3111899623043834),dotstyle); label("$N$", (-4.6442445535803625,7.574032764042567), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]$
Hence, $N,J,M$ are collinear. Moreover, $\angle D'IT=\angle D'DB=\angle D'NJ$ $\implies$ $D'NJI$ is cyclic. Hence, by Radical Axes Theorem, we're done! $\qquad \square$

Taking a look at the Taiwan TST problem, we realise that the steps required to prove it are already mentioned in the Mathematical Reflections problem, hence are work is reduced (lol)!

Lemme continue this in Part (VI).

This post has been edited 33 times. Last edited by AlastorMoody, Nov 19, 2019, 11:09 AM

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Kukuku first shout of 2025
by HoRI_DA_GRe8, 2 hours ago
what a goat, u used to be friends with my brother
by bookstuffthanks, Jul 31, 2024, 12:05 PM
hello fellow moody!!
by crazyeyemoody907, Oct 31, 2023, 1:55 AM
@below I wish I started earlier / didn't have to do JEE and leave oly way before I could study conics and projective stuff which I really wanted to study . Huh, life really sucks when u are forced due to peer pressure to read sh_t u dont want to read
by kamatadu, Jan 3, 2023, 1:25 PM
Lots of good stuffs here.
by amar_04, Dec 30, 2022, 2:31 PM
But even if he went to jee he could continue with this.

Doing JEE(and completely leaving oly) seems like a insult to the oly math he knows
by HoRI_DA_GRe8, Feb 11, 2022, 2:11 PM
Ohhh did he go for JEE? Good for him, bad for us :sadge:. Hmmm so that is the reason why he is inactive
Btw @below finally everyone falls to the monopoly of JEE Coz IIT's are the best in India.
by BVKRB-, Feb 1, 2022, 12:57 PM
Kukuku first shout of 2022,why did this guy left this and went for trashy JEE
by Commander_Anta78, Jan 27, 2022, 3:42 PM
When are you going to br alive again ,we miss you
by HoRI_DA_GRe8, Aug 11, 2021, 5:10 PM
kukuku first shout o 2021
by leafwhisker, Mar 6, 2021, 5:10 AM
wow I completely forgot this blog
by Math-wiz, Dec 25, 2020, 6:49 PM
buuuuuujmmmmpppp
by DuoDuoling0, Dec 22, 2020, 10:54 PM
This site would work faster if not all diagrams were displayed on the initial page. Anyway I like your problem selection taste.
by WolfusA, Sep 24, 2020, 7:58 PM
nice blog
by Orestis_Lignos, Sep 15, 2020, 9:09 AM
Hello everyone, nice blog
by Functional_equation, Sep 12, 2020, 6:22 PM
Nice !
by Synthetic_Potato, Mar 10, 2019, 8:30 AM
Thanks dude!
by AlastorMoody, Mar 1, 2019, 1:39 PM
Amazing work bro!!
by Night_Witch123, Mar 1, 2019, 1:33 PM
@below Thanks a lot
by AlastorMoody, Feb 26, 2019, 2:48 PM
Nice Blog, loved it.
by Delta0001, Feb 26, 2019, 6:07 AM
@below Thanks a lot!
by AlastorMoody, Feb 16, 2019, 3:11 PM
Nice work.
by TheDarkPrince, Feb 16, 2019, 2:39 PM
@below Thanks buddy!
by AlastorMoody, Jan 29, 2019, 9:31 AM
Nice blog! 6th shout!
by PhysicsMonster_01, Jan 29, 2019, 9:21 AM
@below I am noob right now I hope to develop myself within a year or so
by AlastorMoody, Jan 28, 2019, 7:24 PM
Nice blog. But the properties are a bit 'basic'.
by ayan.nmath, Jan 28, 2019, 7:14 PM
Nice!!!!!!!!
by enthusiast101, Jan 13, 2019, 4:44 PM
can I have contrib?
by Kagebaka, Dec 23, 2018, 8:24 PM
Math is fun!
by karolina.newgard, Nov 21, 2018, 6:37 PM

121 shouts

Magic of Hogwarts

Incenter-Related Configurations -Part (V)

by AlastorMoody, Nov 18, 2019, 2:40 AM

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