Incenter-Related Configurations -Part (II) [Feat. Sharky Devil Lemma]

by AlastorMoody, Nov 17, 2019, 10:40 AM

Ok, Let's review! Here's the link to Part (I).
[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(21cm); 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 = -20.310021989506694, xmax = 16.515608576902242, ymin = -13.697015176967454, ymax = 8.18706856126748; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen ttffqq = rgb(0.2,1,0); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); pen ubqqys = rgb(0.29411764705882354,0,0.5098039215686274); pen qqqqcc = rgb(0,0,0.8); draw((-4.8127515399252045,5.283566240744204)--(-7.32,-3.13)--(3.64,-3.27)--cycle, linewidth(2) + rvwvcq); /* draw figures */ draw((-4.8127515399252045,5.283566240744204)--(-7.32,-3.13), linewidth(0.4) + rvwvcq); draw((-7.32,-3.13)--(3.64,-3.27), linewidth(0.4) + rvwvcq); draw((3.64,-3.27)--(-4.8127515399252045,5.283566240744204), linewidth(0.4) + rvwvcq); draw(circle((-1.8016037461609404,-0.1941218423136115), 6.250766174731304), linewidth(0.4)); draw(circle((-1.8814427861656493,-6.444378116967962), 6.368909394110248), linewidth(0.4) + linetype("4 4")); draw((-3.425794343555926,-0.26554447803803993)--(-7.32,-3.13), linewidth(0.4) + ttffqq); draw((-7.32,-3.13)--(-0.33709122877537645,-12.623211755897882), linewidth(0.4) + ttffqq); draw((-0.33709122877537645,-12.623211755897882)--(3.64,-3.27), linewidth(0.4) + ttffqq); draw((3.64,-3.27)--(-3.425794343555926,-0.26554447803803993), linewidth(0.4) + ttffqq); draw((-4.8127515399252045,5.283566240744204)--(-0.33709122877537645,-12.623211755897882), linewidth(0.4)); draw((-7.32,-3.13)--(-1.7217647061562313,6.056134432340744), linewidth(0.4) + dtsfsf); draw((-1.7217647061562313,6.056134432340744)--(3.64,-3.27), linewidth(0.4) + dtsfsf); draw((-1.7217647061562313,6.056134432340744)--(-1.88144278616565,-6.444378116967967), linewidth(0.4)); draw((-1.7217647061562313,6.056134432340744)--(-6.321988149529278,-11.009962827199013), linewidth(0.4) + ubqqys); draw((-7.32,-3.13)--(-6.321988149529278,-11.009962827199013), linewidth(0.4) + ubqqys); draw((-6.321988149529278,-11.009962827199013)--(3.64,-3.27), linewidth(0.4) + ubqqys); draw((-14.51837862789293,-3.03804990803786)--(-3.425794343555926,-0.26554447803803993), linewidth(0.4) + ttffqq); draw((-14.51837862789293,-3.03804990803786)--(-6.321988149529278,-11.009962827199013), linewidth(0.4) + ttffqq); draw((-14.51837862789293,-3.03804990803786)--(-7.32,-3.13), linewidth(0.4) + rvwvcq); draw((-14.51837862789293,-3.03804990803786)--(-1.88144278616565,-6.444378116967967), linewidth(0.4) + linetype("4 4")); draw(circle((-3.425794343555926,-0.26554447803803993), 2.913961301965583), linewidth(0.4) + qqqqcc); draw(circle((-4.119272941740566,2.5090108813530825), 2.8599073426364945), linewidth(0.4)); draw((-6.755455378061811,3.6178896713466084)--(-1.88144278616565,-6.444378116967967), linewidth(0.4) + linetype("4 4")); draw((-14.51837862789293,-3.03804990803786)--(-4.8127515399252045,5.283566240744204), linewidth(0.4) + ttffqq); draw((-6.755455378061811,3.6178896713466084)--(1.209544047603321,-5.67180992537143), linewidth(0.4) + dtsfsf); draw(circle((-3.409081408517464,1.0428395792587513), 4.2224520981067), linewidth(0.4) + ubqqys); draw((-6.755455378061811,3.6178896713466084)--(-1.7217647061562313,6.056134432340744), linewidth(0.4) + ubqqys); draw(circle((-8.97208648572443,-1.6517971930379418), 5.716909402476542), linewidth(0.4) + linetype("4 4")); draw(circle((-9.665565083909069,1.122758166353172), 6.392348795598595), linewidth(0.4) + linetype("4 4")); draw((-4.8127515399252045,5.283566240744204)--(-4.92061560412313,-3.160649070750248), linewidth(0.4)); draw((-3.3551493764184945,5.2649472350059785)--(-3.4630134406164252,-3.1792680764884764), linewidth(0.4)); draw((-4.883396507062627,-0.2469254722998166)--(-3.425794343555926,-0.26554447803803993), linewidth(0.4)); draw(circle((-8.199910707029291,-4.741214012502913), 6.5439899781189395), linewidth(0.4) + linetype("4 4")); draw((-6.218395061190416,0.5666524344894496)--(-1.3531342778462379,1.7826866420666359), linewidth(0.4) + ttffqq); /* dots and labels */ dot((-4.8127515399252045,5.283566240744204),dotstyle); label("$A$", (-4.704406413413728,5.591163455766509), NE * labelscalefactor); dot((-7.32,-3.13),dotstyle); label("$B$", (-7.209756689653024,-2.8304356655680385), NE * labelscalefactor); dot((3.64,-3.27),dotstyle); label("$C$", (3.7473776510079952,-2.9813603810041416), NE * labelscalefactor); dot((-3.425794343555926,-0.26554447803803993),dotstyle); label("$I$", (-3.3158990314015875,0.03713392771791844), NE * labelscalefactor); dot((-0.33709122877537645,-12.623211755897882),dotstyle); label("$I_A$", (-0.20684989341788235,-12.308507794955307), NE * labelscalefactor); dot((-1.88144278616565,-6.444378116967967),dotstyle); label("$M_A$", (-1.7462819908661247,-6.150779405162305), NE * labelscalefactor); dot((-1.7217647061562313,6.056134432340744),dotstyle); label("$M_{BC}$", (-1.5953572754300225,6.345787032947024), NE * labelscalefactor); dot((-4.873891246542604,-5.637753652618531),dotstyle); label("$T$", (-4.764776299588169,-5.335785941807348), NE * labelscalefactor); dot((-6.321988149529278,-11.009962827199013),dotstyle); label("$J$", (-6.21365356777475,-10.708705811332615), NE * labelscalefactor); dot((-14.51837862789293,-3.03804990803786),dotstyle); label("$K$", (-14.393773144411488,-2.739880836306377), NE * labelscalefactor); dot((-3.4630134406164252,-3.1792680764884764),dotstyle); label("$D$", (-3.346083974488808,-2.89080555174248), NE * labelscalefactor); dot((-1.3531342778462379,1.7826866420666359),dotstyle); label("$E$", (-1.2331379583833773,2.089710057648919), NE * labelscalefactor); dot((-6.218395061190416,0.5666524344894496),dotstyle); label("$F$", (-6.092913795425868,0.8823123341600952), NE * labelscalefactor); dot((-6.755455378061811,3.6178896713466084),dotstyle); label("$L$", (-6.636242770995835,3.930991585969376), NE * labelscalefactor); dot((1.209544047603321,-5.67180992537143),dotstyle); label("$A'$", (1.33258220403036,-5.365970884894569), NE * labelscalefactor); dot((-3.3551493764184945,5.2649472350059785),dotstyle); label("$G$", (-3.2253442021399263,5.560978512679289), NE * labelscalefactor); dot((-4.92061560412313,-3.160649070750248),dotstyle); label("$H_A$", (-4.794961242675389,-2.8606206086552595), NE * labelscalefactor); dot((-4.883396507062627,-0.2469254722998166),dotstyle); label("$G'$", (-4.764776299588169,0.06731887080513904), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]
Let's dive in to another configuration! From aops29's blogpost, we can have a good intro in this configuration.
aops29 wrote:
\[\textbf{The Sharky-Devil Lemma}\]In triangle \(ABC\), let \(DEF\) be the contact triangle, and let \(M\) be the midpoint of the arc \(BC\) not containing \(A\) in \((ABC)\). Suppose ray \(MD\) meets \((ABC)\) again at \(T\). If \(I\) is the incenter of \(ABC\) and ray \(TI\) intersects \((ABC)\) again at \(S\), then \(S\) is the antipode of \(A\). If \(P=TS\cap EF\), then \(DP\perp EF\).

[asy] unitsize(100); import olympiad; import cse5; //the config pair A,B,C,I; B=origin; C=2*right; A=1.2*dir(65); I=incenter(A,B,C); dot(A); dot(B); dot(C); dot(I); label("$I$",I,S); label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); draw(A--B--C--cycle); draw(circumcircle(A,B,C)); pair D,E,F; D=foot(I,B,C); E=foot(I,C,A); F=foot(I,A,B); dot(D); dot(E); dot(F); label("$D$",D,S); label("$E$",E,NE); label("$F$",F,NW); draw(D--E--F--cycle); pair P=foot(D,E,F); dot(P); label("$P$",P,N); path w = circumcircle(A,B,C); pair T = OP(w,circumcircle(A,E,F)); dot(T); label("$T$",T,NW); pair M = OP(Line(T,D,10),w); dot(M); label("$M$",M,S); draw(D--P); draw(rightanglemark(D,P,E,2),green); pair S = OP(Line(T,I,10),w); dot(S); label("$S$",S,SE); draw(T--S); draw(A--S,dashed+red); draw(A--T); draw(rightanglemark(A,T,S,2),red); draw(M--T,blue); [/asy]
This is well-known, but since I like the name I mentioned this here :D Now Let's prove this!! Referring to aops29's diagram, we just need to show that If $TS \cap EF=S$, then $DP \perp EF$ coz rest all we have proved earlier :P
We'l work with the original diagram and erase some points to clear mess,
[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(21cm); 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 = -25.06371542459404, xmax = 14.590283440692431, ymin = -11.695057807796225, ymax = 7.388429146123024; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen ttffqq = rgb(0.2,1,0); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); pen qqqqcc = rgb(0,0,0.8); pen ubqqys = rgb(0.29411764705882354,0,0.5098039215686274); draw((-4.8127515399252045,5.283566240744204)--(-7.32,-3.13)--(3.64,-3.27)--cycle, linewidth(2) + rvwvcq); /* draw figures */ draw((-4.8127515399252045,5.283566240744204)--(-7.32,-3.13), linewidth(0.4) + rvwvcq); draw((-7.32,-3.13)--(3.64,-3.27), linewidth(0.4) + rvwvcq); draw((3.64,-3.27)--(-4.8127515399252045,5.283566240744204), linewidth(0.4) + rvwvcq); draw(circle((-1.8016037461609404,-0.1941218423136115), 6.250766174731304), linewidth(0.4)); draw((-3.425794343555926,-0.26554447803803993)--(-7.32,-3.13), linewidth(0.4) + ttffqq); draw((3.64,-3.27)--(-3.425794343555926,-0.26554447803803993), linewidth(0.4) + ttffqq); draw((-7.32,-3.13)--(-1.7217647061562313,6.056134432340744), linewidth(0.4) + dtsfsf); draw((-1.7217647061562313,6.056134432340744)--(3.64,-3.27), linewidth(0.4) + dtsfsf); draw((-1.7217647061562313,6.056134432340744)--(-1.88144278616565,-6.444378116967967), linewidth(0.4)); draw((-14.51837862789293,-3.03804990803786)--(-3.425794343555926,-0.26554447803803993), linewidth(0.4) + ttffqq); draw((-14.51837862789293,-3.03804990803786)--(-7.32,-3.13), linewidth(0.4) + rvwvcq); draw(circle((-3.425794343555926,-0.26554447803803993), 2.913961301965583), linewidth(0.4) + qqqqcc); draw(circle((-4.119272941740566,2.5090108813530825), 2.8599073426364945), linewidth(0.4)); draw((-6.755455378061811,3.6178896713466084)--(-1.88144278616565,-6.444378116967967), linewidth(0.4)); draw((-14.51837862789293,-3.03804990803786)--(-4.8127515399252045,5.283566240744204), linewidth(0.4) + ttffqq); draw(circle((-3.409081408517464,1.0428395792587513), 4.2224520981067), linewidth(0.4) + ubqqys); draw((-6.755455378061811,3.6178896713466084)--(-1.7217647061562313,6.056134432340744), linewidth(0.4) + ubqqys); draw((-3.3551493764184945,5.2649472350059785)--(-3.4630134406164252,-3.1792680764884764), linewidth(0.4)); draw((-6.218395061190416,0.5666524344894496)--(-1.3531342778462379,1.7826866420666359), linewidth(0.4) + ttffqq); draw((-4.8127515399252045,5.283566240744204)--(-1.88144278616565,-6.444378116967967), linewidth(0.4)); draw(circle((-11.902939656253105,-3.0714588000113654), 8.440614750448322), linewidth(0.4) + linetype("4 4")); draw(circle((-8.44211877371571,-10.216230347254088), 13.93656019609169), linewidth(0.4) + linetype("4 4")); draw((-20.342865871889785,-2.9636495235342566)--(-14.51837862789293,-3.03804990803786), linewidth(0.4) + rvwvcq); draw((-20.342865871889785,-2.9636495235342566)--(-6.218395061190416,0.5666524344894496), linewidth(0.4) + ttffqq); draw((-20.342865871889785,-2.9636495235342566)--(-6.755455378061811,3.6178896713466084), linewidth(0.4) + ubqqys); draw((-4.506237629689092,0.994592908428854)--(-3.4630134406164252,-3.1792680764884764), linewidth(0.4) + dtsfsf); draw((-20.342865871889785,-2.9636495235342566)--(-1.88144278616565,-6.444378116967967), linewidth(0.4) + ubqqys); draw((-1.7217647061562313,6.056134432340744)--(-4.003196202301187,-6.044341361975363), linewidth(0.4)); draw((-4.8127515399252045,5.283566240744204)--(-4.003196202301187,-6.044341361975363), linewidth(0.4)); /* dots and labels */ dot((-4.8127515399252045,5.283566240744204),dotstyle); label("$A$", (-4.713503506922719,5.570954198130715), NE * labelscalefactor); dot((-7.32,-3.13),dotstyle); label("$B$", (-7.219415935215128,-2.855520560742719), NE * labelscalefactor); dot((3.64,-3.27),dotstyle); label("$C$", (3.7405087511626602,-2.9932080568027426), NE * labelscalefactor); dot((-3.425794343555926,-0.26554447803803993),dotstyle); label("$I$", (-3.30909104711049,0.008379357305768217), NE * labelscalefactor); dot((-1.88144278616565,-6.444378116967967),dotstyle); label("$M_A$", (-1.7669910912382387,-6.160020466183282), NE * labelscalefactor); dot((-1.7217647061562313,6.056134432340744),dotstyle); label("$M_{BC}$", (-1.6017660959662117,6.342004176066847), NE * labelscalefactor); dot((-14.51837862789293,-3.03804990803786),dotstyle); label("$K$", (-14.4067032295483,-2.7729080631067053), NE * labelscalefactor); dot((-3.4630134406164252,-3.1792680764884764),dotstyle); label("$D$", (-3.3641660455344993,-2.9105955591667287), NE * labelscalefactor); dot((-1.3531342778462379,1.7826866420666359),dotstyle); label("$E$", (-1.2437786062101532,2.0461542989941153), NE * labelscalefactor); dot((-6.218395061190416,0.5666524344894496),dotstyle); label("$F$", (-6.117915966734949,0.8345043336659088), NE * labelscalefactor); dot((-6.755455378061811,3.6178896713466084),dotstyle); label("$L$", (-6.641128451763034,3.8911667461984294), NE * labelscalefactor); dot((1.209544047603321,-5.67180992537143),dotstyle); label("$A'$", (1.3172088205062646,-5.388970488247151), NE * labelscalefactor); dot((-3.3551493764184945,5.2649472350059785),dotstyle); label("$G$", (-3.254016048686481,5.54341669891871), NE * labelscalefactor); dot((-4.506237629689092,0.994592908428854),dotstyle); label("$P$", (-4.3830535163786655,1.2751043210579838), NE * labelscalefactor); dot((-20.342865871889785,-2.9636495235342566),dotstyle); label("$X$", (-20.244653062493253,-2.690295565470691), NE * labelscalefactor); dot((-4.003196202301187,-6.044341361975363),dotstyle); label("$V$", (-3.8873785305625845,-5.774495477215217), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

Let $EF$ $\cap$ $BC$ $=X$. By Radical Axes Theorem, $X$ $\in$ $LG$. Let $P$ be foot from $D$ to $EF$. Hence, $XLPD$ is cyclic. Let this intersect $\odot (ABC)$ $=V$. Now observe, $\Psi _{\odot (BIC)}$ $( \odot (XLPDV))$ $=$ $\odot (XLPDV)$ $\implies$ $V$ $\in$ $XM_A$ and since, $\angle DVX$ $=$ $90^{\circ}$ $\implies$ $V$ $\in$ $DM_{BC}$. Also, By Converse of Reim's Theorem, $V$ $\in$ $AP$. Applying Reim's Theorem again $\implies$ $P$ $\in$ $LA'$, Hence the Sharky-Devil Lemma is proved!

Let $T_A$ be the foot from $I_A$ to $BC$. Let $D'$ be $D-$antipode in $(I)$. It's Well-Known, $D'$ $\in$ $AT_A$. Also $\sqrt{bc}$ Inversion implies, $AT,AT_A$ are Isogonal. Hence, by Reflection Argument, $AT$ intersects $(I)$ at the reflection of Orthocenter of $\Delta DEF$ over $EF$. Let this point be $Q$. The fact that $Q$ $\in$ $\odot (LIDT)$ follows from Simple POP.
[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 = -15.008872990343747, xmax = 12.804220510314202, ymin = -8.92128006748609, ymax = 6.761774282356115; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen ttffqq = rgb(0.2,1,0); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); pen ubqqys = rgb(0.29411764705882354,0,0.5098039215686274); pen qqqqcc = rgb(0,0,0.8); pen ffdxqq = rgb(1,0.8431372549019608,0); draw((-5.942203631855723,4.138673889328595)--(-7.045966190852679,-2.451453892808012)--(3.5400471222335197,-2.7410498081490866)--cycle, linewidth(2) + rvwvcq); /* draw figures */ draw((-5.942203631855723,4.138673889328595)--(-7.045966190852679,-2.451453892808012), linewidth(0.4) + rvwvcq); draw((-7.045966190852679,-2.451453892808012)--(3.5400471222335197,-2.7410498081490866), linewidth(0.4) + rvwvcq); draw((3.5400471222335197,-2.7410498081490866)--(-5.942203631855723,4.138673889328595), linewidth(0.4) + rvwvcq); draw(circle((-1.680910362053616,0.037464498041687326), 5.914265702725966), linewidth(0.4)); draw(circle((-1.8426432646069566,-5.874589398082089), 6.228356618102593), linewidth(0.4) + linetype("4 4")); draw((-4.202494036745758,-0.11060375484023069)--(-7.045966190852679,-2.451453892808012), linewidth(0.4) + ttffqq); draw((3.5400471222335197,-2.7410498081490866)--(-4.202494036745758,-0.11060375484023069), linewidth(0.4) + ttffqq); draw((-7.045966190852679,-2.451453892808012)--(-1.519177459500276,5.949518394165464), linewidth(0.4) + dtsfsf); draw((-1.519177459500276,5.949518394165464)--(3.5400471222335197,-2.7410498081490866), linewidth(0.4) + dtsfsf); draw((-1.519177459500276,5.949518394165464)--(-1.842643264606957,-5.874589398082089), linewidth(0.4)); draw((-1.519177459500276,5.949518394165464)--(-7.696524173231269,-8.001677659040286), linewidth(0.4) + ubqqys); draw((-7.045966190852679,-2.451453892808012)--(-7.696524173231269,-8.001677659040286), linewidth(0.4) + ubqqys); draw((-7.696524173231269,-8.001677659040286)--(3.5400471222335197,-2.7410498081490866), linewidth(0.4) + ubqqys); draw((-9.740054607429487,-2.3777531541131265)--(-4.202494036745758,-0.11060375484023069), linewidth(0.4) + ttffqq); draw((-9.740054607429487,-2.3777531541131265)--(-7.045966190852679,-2.451453892808012), linewidth(0.4) + rvwvcq); draw((-9.740054607429487,-2.3777531541131265)--(-1.842643264606957,-5.874589398082089), linewidth(0.4) + linetype("4 4")); draw(circle((-4.202494036745758,-0.11060375484023069), 2.4177329700687884), linewidth(0.4) + qqqqcc); draw(circle((-5.072348834300742,2.0140350672441834), 2.295808679558832), linewidth(0.4)); draw((-7.349716905373479,1.72363506380737)--(-1.842643264606957,-5.874589398082089), linewidth(0.4)); draw((-9.740054607429487,-2.3777531541131265)--(-5.942203631855723,4.138673889328595), linewidth(0.4) + ttffqq); draw(circle((-4.178123331091742,0.7802534735805249), 3.308923484338092), linewidth(0.4) + ubqqys); draw((-7.349716905373479,1.72363506380737)--(-1.519177459500276,5.949518394165464), linewidth(0.4) + ubqqys); draw(circle((-6.971274322087621,-1.2441784544766825), 2.991844893732568), linewidth(0.4) + linetype("4 4")); draw(circle((-7.841129119642604,0.8804603676077339), 3.7711899131827487), linewidth(0.4) + linetype("4 4")); draw((-4.087636730328725,4.087939492580288)--(-4.2686099318547495,-2.527432545419236), linewidth(0.4)); draw(circle((-5.79134893601822,-4.126171276097608), 4.318476862126188), linewidth(0.4) + linetype("4 4")); draw((-6.587013075226309,0.2887728464791664)--(-2.7826744728952577,1.846318604545808), linewidth(0.4) + ttffqq); draw((-5.942203631855723,4.138673889328595)--(-1.842643264606957,-5.874589398082089), linewidth(0.4)); draw(circle((-8.57911002453525,-2.409512495349541), 4.312112728953992), linewidth(0.4) + linetype("4 4")); draw(circle((-4.8139720411984035,-7.604351967915097), 9.666506301667258), linewidth(0.4) + linetype("4 4")); draw((-12.889610117215755,-2.2915924452798433)--(-9.740054607429487,-2.3777531541131265), linewidth(0.4) + rvwvcq); draw((-12.889610117215755,-2.2915924452798433)--(-6.587013075226309,0.2887728464791664), linewidth(0.4) + ttffqq); draw((-12.889610117215755,-2.2915924452798433)--(-7.349716905373479,1.72363506380737), linewidth(0.4) + ubqqys); draw((-5.588903275433209,0.6974119943720299)--(-4.2686099318547495,-2.527432545419236), linewidth(0.4) + dtsfsf); draw((-12.889610117215755,-2.2915924452798433)--(-1.842643264606957,-5.874589398082089), linewidth(0.4) + ubqqys); draw((-1.519177459500276,5.949518394165464)--(-5.019984567584007,-4.844043696400092), linewidth(0.4)); draw((-5.942203631855723,4.138673889328595)--(-5.019984567584007,-4.844043696400092), linewidth(0.4)); draw((-7.349716905373479,1.72363506380737)--(2.5803829077484957,-4.0637448932452145), linewidth(0.4) + dtsfsf); draw((-6.587013075226309,0.2887728464791664)--(-4.2686099318547495,-2.527432545419236), linewidth(0.4) + ttffqq); draw((-4.2686099318547495,-2.527432545419236)--(-2.7826744728952577,1.846318604545808), linewidth(0.4) + ttffqq); draw((-1.842643264606957,-5.874589398082089)--(0.5172075075318441,-11.638575041323948), linewidth(0.4)); draw((0.7626908632355884,-2.665071155537863)--(0.5172075075318441,-11.638575041323948), linewidth(0.4) + dtsfsf); draw((-5.942203631855723,4.138673889328595)--(0.7626908632355884,-2.665071155537863), linewidth(0.8) + ffdxqq); draw((-5.942203631855723,4.138673889328595)--(-5.949509104988513,-4.0561407069402575), linewidth(0.8) + ffdxqq); draw((-5.944497144381616,1.5659575734578608)--(-5.588903275433209,0.6974119943720299), linewidth(0.4) + linetype("4 4") + dtsfsf); draw((xmin, -0.02735646619517119*xmin + 3.9761161965489102)--(xmax, -0.02735646619517119*xmax + 3.9761161965489102), linewidth(0.4) + linetype("2 2") + rvwvcq); /* line */ draw((-5.949509104988513,-4.0561407069402575)--(2.7982309662885934,3.899566485713355), linewidth(0.4)); /* dots and labels */ dot((-5.942203631855723,4.138673889328595),dotstyle); label("$A$", (-5.843451617059394,4.362923256261925), NE * labelscalefactor); dot((-7.045966190852679,-2.451453892808012),dotstyle); label("$B$", (-6.952354449876513,-2.2226017304683516), NE * labelscalefactor); dot((3.5400471222335197,-2.7410498081490866),dotstyle); label("$C$", (3.63879913702985,-2.5168004412157523), NE * labelscalefactor); dot((-4.202494036745758,-0.11060375484023069),dotstyle); label("$I$", (-4.100890022632492,0.10835728545336126), NE * labelscalefactor); dot((-1.842643264606957,-5.874589398082089),dotstyle); label("$M_A$", (-1.7473003366533006,-5.639832909149698), NE * labelscalefactor); dot((-1.519177459500276,5.949518394165464),dotstyle); label("$M_{BC}$", (-1.4304709558484094,6.173376860861314), NE * labelscalefactor); dot((-5.949509104988513,-4.0561407069402575),dotstyle); label("$T$", (-5.866082287116886,-3.8293793045503093), NE * labelscalefactor); dot((-7.696524173231269,-8.001677659040286),dotstyle); label("$J$", (-7.608643881543788,-7.76711589455398), NE * labelscalefactor); dot((-9.740054607429487,-2.3777531541131265),dotstyle); label("$K$", (-9.645404186718089,-2.154709720295875), NE * labelscalefactor); dot((-4.2686099318547495,-2.527432545419236),dotstyle); label("$D$", (-4.168782032804969,-2.290493740640829), NE * labelscalefactor); dot((-2.7826744728952577,1.846318604545808),dotstyle); label("$E$", (-2.6977884790679743,2.0772255804551967), NE * labelscalefactor); dot((-6.587013075226309,0.2887728464791664),dotstyle); label("$F$", (-6.499741048726668,0.5157093464882238), NE * labelscalefactor); dot((-7.349716905373479,1.72363506380737),dotstyle); label("$L$", (-7.2691838306814045,1.9414415601102424), NE * labelscalefactor); dot((2.5803829077484957,-4.0637448932452145),dotstyle); label("$A'$", (2.665680324557684,-3.8293793045503093), NE * labelscalefactor); dot((-4.087636730328725,4.087939492580288),dotstyle); label("$G$", (-3.987736672345031,4.31766191614694), NE * labelscalefactor); dot((-5.588903275433209,0.6974119943720299),dotstyle); label("$P$", (-5.50399156619701,0.9230614075230862), NE * labelscalefactor); dot((-12.889610117215755,-2.2915924452798433),dotstyle); label("$X$", (-12.791067324709507,-2.0641870400659053), NE * labelscalefactor); dot((-5.019984567584007,-4.844043696400092),dotstyle); label("$V$", (-4.938224814759705,-4.621452756562542), NE * labelscalefactor); dot((0.7626908632355884,-2.665071155537863),dotstyle); label("$T_A$", (0.8552267199583057,-2.4489084310432756), NE * labelscalefactor); dot((-4.136378141636767,2.306225035738775),dotstyle); label("$D'$", (-4.055628682517508,2.5298389816050437), NE * labelscalefactor); dot((-5.944497144381616,1.5659575734578608),dotstyle); label("$Q$", (-5.843451617059394,1.783026869707796), NE * labelscalefactor); dot((2.7982309662885934,3.899566485713355),dotstyle); label("$A_1$", (2.891987025132606,4.136616555687001), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

Now We will focus on the Involvement of the $A-$antipode in this diagram. Clear some unnecessary lines & points. Define: $R$ $=$ $AL$ $\cap$ $EF$. Let $M'$ be midpoint of $EF$ $\implies$ $RLM'I$ is cyclic. Let $RA'$ $\cap \odot (ABC)$ $=$ $Y$ $\implies$ $RYM'A$ is cyclic. By POP, $AFEA'$ is also cyclic. We also know, $P$ is orthocenter WRT $\Delta ARI$. Let $R'$ be foot from $A$ to $RI$, then a series of Power of Point follows, $$RP \times RM'=AP \times PR'=LP \times PI = FP \times PE$$Thus, $P$ lies on Radical Axes of $\odot (A'EF)$ and $\odot (ARY)$. Pretty Cool, isn't it? :) (You said no? :mad: Well, this trivialises ELMO SL 2019 G3, so well, yeah this is pretty interesting!!)
[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(20cm); 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 = -15.635835279685946, xmax = 8.193920498944122, ymin = -6.769346326188248, ymax = 6.667611163307858; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen ttffqq = rgb(0.2,1,0); pen qqqqcc = rgb(0,0,0.8); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); draw((-4.446753459113518,4.611610926456197)--(-7.045966190852679,-2.451453892808012)--(3.5400471222335197,-2.7410498081490866)--cycle, linewidth(2) + rvwvcq); /* draw figures */ draw((-4.446753459113518,4.611610926456197)--(-7.045966190852679,-2.451453892808012), linewidth(0.4) + rvwvcq); draw((-7.045966190852679,-2.451453892808012)--(3.5400471222335197,-2.7410498081490866), linewidth(0.4) + rvwvcq); draw((3.5400471222335197,-2.7410498081490866)--(-4.446753459113518,4.611610926456197), linewidth(0.4) + rvwvcq); draw(circle((-1.6931921700442931,-0.4114899795963873), 5.728319342097372), linewidth(0.4)); draw((-3.345933085153599,0.055038275929575345)--(-7.045966190852679,-2.451453892808012), linewidth(0.4) + ttffqq); draw((3.5400471222335197,-2.7410498081490866)--(-3.345933085153599,0.055038275929575345), linewidth(0.4) + ttffqq); draw((-1.536544200453946,5.314687095789082)--(-1.8498401396346391,-6.137667054981856), linewidth(0.4)); draw((-13.041975271686374,-2.2874242730822445)--(-3.345933085153599,0.055038275929575345), linewidth(0.4) + ttffqq); draw((-13.041975271686374,-2.2874242730822445)--(-7.045966190852679,-2.451453892808012), linewidth(0.4) + rvwvcq); draw(circle((-3.345933085153599,0.055038275929575345), 2.6067367718043304), linewidth(0.4) + qqqqcc); draw(circle((-3.8963432721335582,2.333324601192886), 2.343830188775013), linewidth(0.4)); draw((-6.001587925321318,3.36360849606723)--(-1.8498401396346391,-6.137667054981856), linewidth(0.4)); draw((-13.041975271686374,-2.2874242730822445)--(-4.446753459113518,4.611610926456197), linewidth(0.4) + ttffqq); draw((-5.792280011836994,0.9552956345685196)--(-1.5804002441062543,1.9728418072278557), linewidth(0.4) + ttffqq); draw((-4.446753459113518,4.611610926456197)--(-1.8498401396346391,-6.137667054981856), linewidth(0.4)); draw((-18.58683965454231,-2.1357363780358547)--(-13.041975271686374,-2.2874242730822445), linewidth(0.4) + rvwvcq); draw((-18.58683965454231,-2.1357363780358547)--(-5.792280011836994,0.9552956345685196), linewidth(0.4) + ttffqq); draw((-4.348502014831068,1.3040973231349209)--(-3.417217522730037,-2.550723633078282), linewidth(0.4) + dtsfsf); draw((-6.001587925321318,3.36360849606723)--(1.0603691190249185,-5.43459088564898), linewidth(0.4) + dtsfsf); draw((-5.792280011836994,0.9552956345685196)--(-3.417217522730037,-2.550723633078282), linewidth(0.4) + ttffqq); draw((-3.417217522730037,-2.550723633078282)--(-1.5804002441062543,1.9728418072278557), linewidth(0.4) + ttffqq); draw((-4.598972421296053,2.3408573894108984)--(-4.348502014831068,1.3040973231349209), linewidth(0.4) + dtsfsf); draw((-10.384072030799011,-0.15403337105498183)--(1.0603691190249185,-5.43459088564898), linewidth(0.4)); draw(circle((-6.865002557976306,-0.04949754756269626), 3.520621776468737), linewidth(0.4) + linetype("2 2")); draw(circle((-7.415412744956264,2.2287887777006015), 3.8066756294725907), linewidth(0.4) + linetype("2 2")); draw(circle((-2.7039828514710456,-2.6021553471088694), 4.710948613364595), linewidth(0.4) + linetype("2 2")); draw((-10.384072030799011,-0.15403337105498183)--(-3.345933085153599,0.055038275929575345), linewidth(0.4) + ttffqq); /* dots and labels */ dot((-4.446753459113518,4.611610926456197),dotstyle); label("$A$", (-4.370507283441791,4.806214454806233), NE * labelscalefactor); dot((-7.045966190852679,-2.451453892808012),dotstyle); label("$B$", (-6.968706855725297,-2.2515813982624286), NE * labelscalefactor); dot((3.5400471222335197,-2.7410498081490866),dotstyle); label("$C$", (3.617986923877646,-2.5424246339658074), NE * labelscalefactor); dot((-3.345933085153599,0.055038275929575345),dotstyle); label("$I$", (-3.2653029877689557,0.2496704287866301), NE * labelscalefactor); dot((-1.8498401396346391,-6.137667054981856),dotstyle); label("$M_A$", (-1.7723077111582843,-5.9355957171718945), NE * labelscalefactor); dot((-1.536544200453946,5.314687095789082),dotstyle); label("$M_{BC}$", (-1.4620749264080148,5.504238220494342), NE * labelscalefactor); dot((-13.041975271686374,-2.2874242730822445),dotstyle); label("$K$", (-12.960077511214873,-2.096465005887293), NE * labelscalefactor); dot((-3.417217522730037,-2.550723633078282),dotstyle); label("$D$", (-3.342861183956523,-2.3485291434968882), NE * labelscalefactor); dot((-1.5804002441062543,1.9728418072278557),dotstyle); label("$E$", (-1.5008540245017985,2.1692357844289307), NE * labelscalefactor); dot((-5.792280011836994,0.9552956345685196),dotstyle); label("$F$", (-5.708386167677327,1.1415896849436586), NE * labelscalefactor); dot((-6.001587925321318,3.36360849606723),dotstyle); label("$L$", (-5.921671207193137,3.5652833158051496), NE * labelscalefactor); dot((1.0603691190249185,-5.43459088564898),dotstyle); label("$A'$", (1.1361246458754912,-5.237571951483785), NE * labelscalefactor); dot((-4.348502014831068,1.3040973231349209),dotstyle); label("$P$", (-4.273559538207331,1.4906015677877134), NE * labelscalefactor); dot((-4.598972421296053,2.3408573894108984),dotstyle); label("$Q$", (-4.5256236758169255,2.5376372163198773), NE * labelscalefactor); dot((-10.384072030799011,-0.15403337105498183),dotstyle); label("$R$", (-10.303709291790693,0.03638538927081889), NE * labelscalefactor); dot((-3.686340127971625,1.4640687208981875),dotstyle); label("$M'$", (-3.6143148706130086,1.6651075092097407), NE * labelscalefactor); dot((-7.301856004362389,-1.5761927222131948),dotstyle); label("$Y$", (-7.220770993334891,-1.3790516911522919), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]
So what did we basically do in this post? We opened a way for dealing with foots on incircle chord and tried involving $A-$antipode. Also, this is what ELMO SL 2019 G3 is based upon. Also, I guess I will soon run outta post-wordlimit, so I'll have to continue this in Part (III) over here
This post has been edited 23 times. Last edited by AlastorMoody, Nov 17, 2019, 12:42 PM

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4 Comments

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Can you show how $Q \odot (LIDT)$?

by hellomath010118, Dec 1, 2019, 1:55 PM

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By POP, $$PQ \times PD=PF \times PE = PI \times PL$$Hence, $Q$ lies on $\odot (LID)$

by AlastorMoody, Dec 4, 2019, 6:36 AM

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This is the most complicated diagram I have seen in my life

by Shauryajain123, Oct 20, 2020, 10:27 AM

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Really cool!!! :coolspeak: :thumbup:

Also, maybe i am partially blind and am telling a fact that was already mentioned somewhere which i missed.

Let $T = ID \cap AC$ and $M_B$ be the mid pt. of the minor arc $AC$. Then apparently, $\overline{L - T - M_B}$ are collinear.

by kamatadu, Oct 4, 2022, 3:46 PM

I'll talk about all possible non-sense :D

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AlastorMoody
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  • 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

  • pro blogo

    by Aritra12, Sep 8, 2020, 11:17 AM

120 shouts
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  • Posts: 2125
  • Joined: Oct 11, 2017
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  • Blog created: Nov 19, 2018
  • Total entries: 25
  • Total visits: 12172
  • Total comments: 60
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