Define $T$ instead as the foot to $EF$ from $D$; we wish to show $T \in GH$. Let $(AI)$ meet $(ABC)$ a second time at a point $T'$ so that $I, T, T'$ are collinear, say by inversion about the incircle. By radical axis on $(AI), (ABC), (A'EFG)$ we get a point $X = AT' \cap EF \cap A'G$. Since $\angle XGA = \angle XMA = 90^{\circ}$, point $X$ lies on $(AMG)$. Now note that \[-1 = (A, I; E, F) \stackrel{T'}{=} (X, T; E, F),\]so by properties of harmonic divisions we have $TM \cdot TX = TE \cdot TF$. This implies that $T$ lies on the radical axis of $(AMG)$ and $(A'EFG)$, as desired.

This is a really rich configuration! tastymath75025 wrote: Let $\triangle ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. The incircle touches sides $BC,CA,$ and $AB$ at $D,E,$ and $F$ respectively, and $A'$ is the reflection of $A$ over $O$. The circumcircles of $ABC$ and $A'EF$ meet at $G$, and the circumcircles of $AMG$ and $A'EF$ meet at a point $H\neq G$, where $M$ is the midpoint of $EF$. Prove that if $GH$ and $EF$ meet at $T$, then $DT\perp EF$. Proposed by Ankit Bisain Let $X=(AMG) \cap AT.$ Since $T$ lies on the radical axis of $(AMG),(EFG),$ hence power of a point gives $X \in (AFE).$ Define $Y=(AEF) \cap (ABC).$ Clearly $\measuredangle IYA=\pi/2=\measuredangle A'YA,$ and so $Y \in A'I.$ Now define $P$ to be the radical center of $(AEF), (FEG), (ABC).$ Hence $P$ lies on $AY,EF$ and $GA'.$ Key Claim: $I,T$ and $Y$ are collinear. Proof: We have $$\measuredangle PMA=\pi/2=\measuredangle A'GA=\measuredangle PGA$$so $P \in (AMG).$ Further, we get $\measuredangle PXA=\pi/2=\measuredangle IXA$ and so $I,X,P$ are also collinear. $\square$ Since $AT \perp PI, PT \perp AI,$ hence $T$ is the orthocenter of $\triangle API.$ Hence $IT \perp AP$ which implies that $T, I, Y$ are collinear. Notice that the power of $I$ with respect to $(AMP)$ is $ $ $IM \cdot IA=r^2,$ where $r$ is the inradius of $ABC.$ So inverting about the incircle of $\triangle ABC,$ we find that $T=AX \cap FE \mapsto (IMP) \cap (IEF)=Y.$ But $Y \in (ABC),$ which is the image of the nine-point circle of $DEF$ under this inversion. So $T$ must be the foot of the perpendicular from $D,$ and so we are done. $\blacksquare$ [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(15cm); 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 = -14.261007052484016, xmax = 13.427339353659017, ymin = -10.769656125630911, ymax = 7.171127796903298; /* image dimensions */ pen zzttff = rgb(0.6,0.2,1); pen ccqqqq = rgb(0.2,0,0); draw((-3.6121766694775372,3.8402861529741235)--(-5.87,-5.79)--(6.97,-5.67)--cycle, linewidth(0) + ccqqqq); /* draw figures */ draw(circle((0.5170807117156297,-2.207636153572406), 7.323123018641797), linewidth(0.2) + blue); draw(circle((-1.649252980059181,-2.412125033326211), 3.338282946745299), linewidth(0.4) + blue); draw(circle((-0.6253105070707227,-5.673642231754901), 5.869971922586647), linewidth(0.2)); 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draw((-1.6180554364422242,-5.750262200340581)--(0.5821675789741579,0.07079569059024113), linewidth(0.2)); draw((-5.821271838735171,1.460253933705996)--(4.646338092908797,-8.255558460118936), linewidth(0.4) + linetype("4 4")); draw((-2.926570903626123,-2.549184361059153)--(-3.6121766694775372,3.8402861529741235), linewidth(0.4)); draw((-3.6121766694775372,3.8402861529741235)--(-10.274577889649324,-3.3376434275920284), linewidth(0.4)); draw((-10.274577889649324,-3.3376434275920284)--(-1.649252980059181,-2.412125033326211), linewidth(0.4) + linetype("4 4")); /* dots and labels */ dot((-3.6121766694775372,3.8402861529741235),dotstyle); label("$A$", (-3.9338597725524567,4.352238105697483), NE * labelscalefactor); dot((-5.87,-5.79),dotstyle); label("$B$", (-6.252198957843215,-6.396425389741515), NE * labelscalefactor); dot((6.97,-5.67),dotstyle); label("$C$", (7.183630411455497,-6.317391099333876), NE * labelscalefactor); dot((0.5170807117156297,-2.207636153572406),linewidth(4pt) + dotstyle); 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label("$M$", (-2.063381566238322,-0.5742326630453913), NE * labelscalefactor); dot((-2.926570903626123,-2.549184361059153),linewidth(4pt) + dotstyle); label("$X$", (-2.985448287660783,-3.1033299560898615), NE * labelscalefactor); dot((-10.274577889649324,-3.3376434275920284),linewidth(4pt) + dotstyle); label("$P$", (-10.54639540332496,-4.078086204450751), NE * labelscalefactor); dot((-5.821271838735171,1.460253933705996),linewidth(4pt) + dotstyle); label("$Y$", (-6.331233248250855,1.5860379414300936), NE * labelscalefactor); dot((-2.052141797423591,0.020277581592099976),linewidth(4pt) + dotstyle); label("$H$", (-1.9580025123614697,0.2424550045002188), NE * labelscalefactor); dot((-3.084191616558338,-1.0802455804311601),linewidth(4pt) + dotstyle); label("$T$", (-3.0644825780684224,-0.6796117169222443), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]

Let $N$ be the middle of the arc $BC$. Claim1: The intersection of $A'I$ and $ND$ lies on $\odot O$. Proof: Let $J = ND \cap \odot O$, then $ND \cdot NJ = NC^2 = NI^2$, thus $\triangle NID \sim \triangle NJI$. Hence \[\angle IJN = \angle NID = 90^\circ - (\angle B + \frac{1}{2} \angle A) =90^\circ - \angle NBA = 90^\circ - \angle NA'A = \angle A'AN = \angle A'JN \]Therefore, $A',I,J$ are collinear. Claim 2: Let $T = A'J \cap EF$, then $DT \perp EF$. Proof: Since $90^\circ = \angle IJA = \angle IMT$, we have $IT \cdot IJ = IM \cdot IA = r^2 = ID^2$. Therefore, \[ \angle TDI = \angle IJD =\angle NID\]implying that $DT \parallel NI$, hence $DT \perp EF$. Claim 3: $AJ$, $EF$, $A'G$ are concurrent, denote the intersection by $L$. Proof: Application of radical axis theorem to $\odot O$, $\odot (AEFIJ)$ and $\odot(A'EFG)$. Claim 4: $L,G,M,A$ are concyclic; $L,I,M,J$ are concyclic. Proof: Since $\angle AML =\angle AGL=90^\circ$ and $\angle IML=\angle IJL =90^\circ$ as well. Finally, $MT \cdot TL = IT \cdot TJ = FT \cdot TE$, meaning that $T$ lies on the radical axis of $\odot(AMLG)$ and $\odot(A'EFG)$, which is $HG$.

Solution. Redefine $T$ as the $D$-foot of altitude in $\bigtriangleup DEF$. It's not hard to show that $T,\ I$ and $A'$ are collinear. Redefine also $H$ as $\overline{GT}\cap (A'EF)$, $G\neq H$, so it suffices to show that $A,\ H,\ M$ and $G$ are concyclic. Let $R=\overline{A'I}\cap (ABC),\ R\neq A'$. Clearly, it lies on $(AEF)$. By the radical axis theorem, $AR,\ EF$ and $GA'$ concur at a point, say $P$. Moreover, being $\angle AMP=\angle AGP=90^\circ$, we infer that $AMGP$ is cyclic. Because $\angle TMI=\angle ART=90^\circ$, we get $$PT\cdot PM=PR\cdot PA=PF\cdot PE$$which gives us that $(P,T;F,E)=-1$, implying the following equality $$PT\cdot TM=FT\cdot TE=GT\cdot TH$$thus $H$ lies on $(PGM)$ and then it lies on $(AMG)$ as well, as required. $\blacksquare$

Here i present a solution that I,mueller.25,amar_04 found. [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(15cm); 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 = -24.620625249953825, xmax = 27.856252010786847, ymin = -16.24019088506529, ymax = 18.41499364361862; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen wvvxds = rgb(0.396078431372549,0.3411764705882353,0.8235294117647058); pen ffqqff = rgb(1,0,1); pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); pen qqffff = rgb(0,1,1); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); pen dbwrru = rgb(0.8588235294117647,0.3803921568627451,0.0784313725490196); /* draw figures */ draw((-3.527646662682087,13.712520226747058)--(-7.26,-4.11), linewidth(0.4) + rvwvcq); draw((-7.26,-4.11)--(13.395645203182287,-4.419894946997829), linewidth(0.4) + rvwvcq); draw((13.395645203182287,-4.419894946997829)--(-3.527646662682087,13.712520226747058), linewidth(0.4) + rvwvcq); draw(circle((-0.14166328775828607,1.5839579345872608), 5.800101026607598), linewidth(0.4) + wvvxds); draw((-5.818616292086269,2.7728130584379342)--(-0.14166328775828446,1.5839579345872612), linewidth(0.4) + ffqqff); draw((-0.14166328775828446,1.5839579345872612)--(4.098565608016035,5.541435771580792), linewidth(0.4) + green); draw((-0.14166328775828446,1.5839579345872612)--(-0.22867193493617494,-4.2154904369538455), linewidth(0.4) + wrwrwr); draw(circle((3.176913954878332,3.006394992923919), 12.632191044978963), linewidth(0.4)); draw((-3.527646662682087,13.712520226747058)--(9.88147457243875,-7.699730240899219), linewidth(0.4) + qqffff); draw((-5.818616292086269,2.7728130584379342)--(4.098565608016035,5.541435771580792), linewidth(0.4) + linetype("2 2") + wvvxds); draw(circle((1.2960492996280217,-3.5659157486317214), 9.528795798604248), linewidth(0.4) + dtsfsf); draw(circle((-9.961397440505575,6.766318481782098), 9.467991748670812), linewidth(0.4) + dtsfsf); draw((-0.5445625005599162,5.78342099642709)--(-8.17647769874491,-2.531903351900492), linewidth(0.4) + ffqqff); draw((-2.442737265572527,3.7152718581028)--(-0.22867193493617494,-4.2154904369538455), linewidth(0.4) + qqffff); draw((-0.14166328775828446,1.5839579345872612)--(-5.12964304175612,-1.3759795466042293), linewidth(0.4) + green); draw((-0.14166328775828446,1.5839579345872612)--(-0.5128097549578929,7.372172013104032), linewidth(0.4) + wrwrwr); draw((-5.12964304175612,-1.3759795466042293)--(-0.5128097549578929,7.372172013104032), linewidth(0.4) + ffqqff); draw((-3.527646662682087,13.712520226747058)--(-0.14166328775828446,1.5839579345872612), linewidth(0.4) + wrwrwr); draw((9.88147457243875,-7.699730240899219)--(-2.442737265572527,3.7152718581028), linewidth(0.4) + blue); draw((-16.395148218329062,-0.17988326318285885)--(-3.527646662682087,13.712520226747058), linewidth(0.4) + linetype("4 4") + wvvxds); draw(circle((-1.8346549752201853,7.64823908066716), 6.296167617885918), linewidth(0.4)); draw((-5.818616292086269,2.7728130584379342)--(-0.5128097549578929,7.372172013104032), linewidth(0.4) + wrwrwr); draw((-0.5128097549578929,7.372172013104032)--(4.098565608016035,5.541435771580792), linewidth(0.4) + wrwrwr); draw((-8.010663903216715,8.872428769327545)--(-5.818616292086269,2.7728130584379342), linewidth(0.4) + blue); draw((-8.010663903216715,8.872428769327545)--(4.098565608016035,5.541435771580792), linewidth(0.4) + blue); draw((-16.395148218329062,-0.17988326318285885)--(9.88147457243875,-7.699730240899219), linewidth(0.4) + linetype("4 4") + wvvxds); draw((-3.527646662682087,13.712520226747058)--(-8.17647769874491,-2.531903351900492), linewidth(0.4) + ffqqff); draw((-5.818616292086269,2.7728130584379342)--(-0.22867193493617494,-4.2154904369538455), linewidth(0.4) + dbwrru); draw((-0.22867193493617494,-4.2154904369538455)--(4.098565608016035,5.541435771580792), linewidth(0.4) + dbwrru); draw((-8.010663903216715,8.872428769327545)--(-2.442737265572527,3.7152718581028), linewidth(0.4) + blue); draw((-16.395148218329062,-0.17988326318285885)--(-5.818616292086269,2.7728130584379342), linewidth(0.4) + linetype("4 4") + wvvxds); /* dots and labels */ dot((-3.527646662682087,13.712520226747058),dotstyle); label("$A$", (-3.391343085381462,14.053279169998621), NE * labelscalefactor); dot((-7.26,-4.11),dotstyle); label("$B$", (-7.139691461148653,-3.768413562058098), NE * labelscalefactor); dot((13.395645203182287,-4.419894946997829),dotstyle); label("$C$", (13.71475586584699,-4.722538603162473), NE * labelscalefactor); dot((-0.14166328775828446,1.5839579345872612),linewidth(4pt) + dotstyle); label("$I$", (-0.017829547190990128,1.8541090015926833), NE * labelscalefactor); dot((-0.22867193493617494,-4.2154904369538455),linewidth(4pt) + dotstyle); label("$D$", (-0.08598133584130269,-3.9387930336838792), NE * labelscalefactor); dot((4.098565608016035,5.541435771580792),linewidth(4pt) + dotstyle); label("$E$", (4.2416572434535444,5.806912743310808), NE * labelscalefactor); dot((-5.818616292086269,2.7728130584379342),linewidth(4pt) + dotstyle); label("$F$", (-5.6744280051669325,3.046765302973152), NE * labelscalefactor); dot((3.1769139548783314,3.0063949929239193),linewidth(4pt) + dotstyle); label("$O$", (3.3216080966743253,3.2852965632492457), NE * labelscalefactor); dot((9.88147457243875,-7.699730240899219),dotstyle); label("$A'$", (10.034559278730113,-7.346382466199505), NE * labelscalefactor); dot((-8.17647769874491,-2.531903351900492),linewidth(4pt) + dotstyle); label("$G$", (-8.025664713602715,-2.269074211751223), NE * labelscalefactor); dot((-0.8600253420351174,4.157124415009363),linewidth(4pt) + dotstyle); label("$M$", (-0.733423328019272,4.443876970304559), NE * labelscalefactor); dot((-0.5445625005599162,5.78342099642709),linewidth(4pt) + dotstyle); label("$H$", (-0.3926643847677092,6.045444003586902), NE * labelscalefactor); dot((-2.442737265572527,3.7152718581028),linewidth(4pt) + dotstyle); label("$T$", (-2.3009144669764607,4.000890344077527), NE * labelscalefactor); dot((-5.12964304175612,-1.3759795466042293),linewidth(4pt) + dotstyle); label("$K$", (-4.992910118663807,-1.1104938046959105), NE * labelscalefactor); dot((-0.5128097549578929,7.372172013104032),linewidth(4pt) + dotstyle); label("$L$", (-0.3926643847677092,7.647011036869246), NE * labelscalefactor); dot((-16.395148218329062,-0.17988326318285885),linewidth(4pt) + dotstyle); label("$T'$", (-16.272031140290537,0.08216249668455824), NE * labelscalefactor); dot((-8.010663903216715,8.872428769327545),linewidth(4pt) + dotstyle); label("$J$", (-7.889361136302091,9.14635038717612), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy] Proof: Let $J=A'I \cap \odot(ABC)$. Notice that it is sufficient to show that if $T$ is the foot of the altitude from $D$ onto $EF$ then $T \in$ radical axis of $\odot(AMG)$ and $\odot(A'EF)$. Now we state a lemma. Claim: If $AJ \cap EF=T'$ then $T'$ is the harmonic conjugate of $T$ w.r.t $EF$. Proof: Firstly it's a well known fact that $\overline{(I,T,J)}$ is a collinear triple (see here ) Notice that since $\overline{IE}=\overline{IF} \implies \angle FJI=\angle IJE$. But also notice that $\angle TJT'=90^\circ$ $\implies$ $(T,T';F,E)=-1$. Done $\square$. Now back to the main problem. Firstly notice that by radical axis theorem on $\odot(ABC),\odot(AEF),\odot(A'EF) \implies AJ,EF,A'G$ are concurrent. So we could define $T'=EF \cap A'G$. But notice that $\angle AMT'=90^\circ$ and also $\angle AGT'=90^\circ$ $\implies$ $T' \in \odot(AMG)$. But now finally notice that $$\text{Pow}_{\odot(A'FE)}{T}=\overline{TF} \cdot \overline{TE}=\overline{TT'} \cdot \overline{TM}=\text{Pow}_{\odot(AMG)}{T}$$where the last part follows from the claim. This immediately implies $T \in $ radical axis of $\odot(A'FE)$ and $\odot(AMG)$ as desired $\blacksquare$.

Comparing to imo problems ,what level is this.Can anyone give his opinion.

Maxito12345 wrote: Comparing to imo problems ,what level is this.Can anyone give his opinion. I would say $\leq$ G4. Actually this is quite a well known configuration now (just a mix of well known lemmas), so it is easy to most of the people. @below Hmm maybe.

AmirKhusrau wrote: Maxito12345 wrote: Comparing to imo problems ,what level is this.Can anyone give his opinion. I would say $\leq$ G4. Actually this is quite a well known configuration now (just a mix of well known lemmas), so it is easy to most of the people. Could it be a p2 (like the one of imo 2019)

Let circumcircle of $AEF$ cut $(ABC)$ at $R$. Claim:$AR,BG,EF$ concur at a point. proof: Radical axis theorem on $(ARFE),(EFA'),(ABC)$ shows that these 3 lines are concurrent.Let this point of concurrency be $S$.$\blacksquare$ Claim:$S$ lies on $AMG$ proof: $\angle AMS= \angle AGS=90^\circ$$\blacksquare$ Now observe that,$ST.TM=GT.TH=FT.TE$. As $M$ is the midpoint of $EF$ we have $(S,T;F,,E)=-1$.[It can be seen considering a circle with diameter $EF$ and centre $M$ then under inversion in this circle $S,T$ swaps, as $ST.TM=FT.TE$].So we have $ST.SM=SF.SE$.From this we get the ninepoint circle of $DEF$ cut $EF$ in $T$ except $M$.So $DT\perp EF$$\blacksquare$

Maxito12345 wrote: AmirKhusrau wrote: Maxito12345 wrote: Comparing to imo problems ,what level is this.Can anyone give his opinion. I would say $\leq$ G4. Actually this is quite a well known configuration now (just a mix of well known lemmas), so it is easy to most of the people. Could it be a p2 (like the one of imo 2019) ?

i think its a medium problem, so sure.

Mr Evan chen ,how many mohs?

bump????

My solution is the same as probably half of this thread, but whatever. Let $T'$ be the foot of the perpendicular from $D$ to $EF$. Let $R = EF \cap (AMG)$, and let $Q = (AEIF) \cap (ABC)$. It suffices to show that $G,T',H$ are collinear, or by radical axes and harmonic bundles it suffices to show that $(E,F;R,T') = -1$. Claim: $Q,T',I$ collinear. Proof: We invert around the incircle. let $Q'$ be the intersection of $T'I$ with $(ABC)$. Note that after the inversion, $Q'$ gets sent to the intersection of line $EF$ with the nine-point circle of $(DEF)$, so $T'$ and $Q'$ are inverses. Now $\angle AQ'I = \angle AMT' = 90$, so $Q=Q'$. Now, note that $\angle RGA + \angle AGA' = 90+90=180$, so $R,G,A'$ are collinear. Moreover, by radical axes on $(AEIF), (A'GFE), (ABA'C)$ we find that $A,Q,R$ are collinear. Now, $(E,F;R,T') \stackrel{Q}{=} (E,F;A,I) = -1$, which is what we wanted.

check this https://artofproblemsolving.com/community/c946900h1911664_properties_of_the_sharkydevil_point

[asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(12cm); 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 = -8.342636385018698, xmax = 22.10453829563805, ymin = -19.202391801926474, ymax = 9.571861192979963; /* image dimensions */ pen ttqqqq = rgb(0.2,0,0); pen wvvxds = rgb(0.396078431372549,0.3411764705882353,0.8235294117647058); pen wwccff = rgb(0.4,0.8,1); pen ttffcc = rgb(0.2,1,0.8); pen ccwwff = rgb(0.8,0.4,1); pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); /* draw figures */ draw((0.36318383958057093,4.154887985795261)--(-0.6571422088705283,-5.117548735216507), linewidth(0.7) + ttqqqq); draw((-0.6571422088705283,-5.117548735216507)--(8.68,-4.66), linewidth(0.7) + ttqqqq); draw((8.68,-4.66)--(0.36318383958057093,4.154887985795261), linewidth(0.7) + ttqqqq); 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draw((2.2525659983910864,-0.16167426106806954)--(-1.112606871601543,-4.572290289023016), linewidth(0.7) + wrwrwr); draw((-1.8933890982991697,1.329770657951845)--(7.270464769693989,-5.98989158807013), linewidth(0.7) + linetype("4 4") + ccwwff); draw((-1.8933890982991697,1.329770657951845)--(4.11717186677921,-7.0466585093944225), linewidth(0.7) + ccwwff); draw((0.36318383958057093,4.154887985795261)--(4.11717186677921,-7.0466585093944225), linewidth(0.7) + ccwwff); draw((-5.953777845649177,-3.7536345426339013)--(0.36318383958057093,4.154887985795261), linewidth(0.7) + wwccff); draw((-5.953777845649177,-3.7536345426339013)--(-0.2985074562530282,-1.858376772832902), linewidth(0.7) + wwccff); draw((-5.953777845649177,-3.7536345426339013)--(7.270464769693989,-5.98989158807013), linewidth(0.7) + wwccff); draw((0.36318383958057093,4.154887985795261)--(-1.112606871601543,-4.572290289023016), linewidth(0.7) + ccwwff); /* dots and labels */ dot((0.36318383958057093,4.154887985795261),dotstyle); label("$A$", (0.49039011574325947,4.486179268298825), NE * labelscalefactor); dot((-0.6571422088705283,-5.117548735216507),dotstyle); label("$B$", (-0.513362895706963,-4.781806870758248), NE * labelscalefactor); dot((8.68,-4.66),dotstyle); label("$C$", (8.821540110780106,-4.313388798748144), NE * labelscalefactor); dot((2.4809144209628675,-2.1642204929126425),linewidth(4pt) + dotstyle); label("$I$", (2.5982714397887268,-1.9043815712676047), NE * labelscalefactor); dot((2.617772544652193,-4.957067822545749),linewidth(4pt) + dotstyle); label("$D$", (2.765563608363764,-4.681431569613226), NE * labelscalefactor); dot((4.514749641394138,-0.2453039629948295),linewidth(4pt) + dotstyle); label("$E$", (4.6392358964041795,0.036207584202829476), NE * labelscalefactor); dot((-0.2985074562530282,-1.858376772832902),linewidth(4pt) + dotstyle); label("$F$", (-0.1787785585568889,-1.6032556678325374), NE * labelscalefactor); dot((3.8168243046372794,-0.9175018011374337),linewidth(4pt) + dotstyle); label("$O$", (3.9366087883890235,-0.6664195238123277), NE * labelscalefactor); dot((7.270464769693989,-5.98989158807013),linewidth(4pt) + dotstyle); 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label("$J$", (-5.833253856393142,-3.4769279558729567), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]

Redefine $T$ be the point on $\overline{EF}$ such that $\overline{DT} \perp \overline{EF}$. Let $L = \overline{AT} \cap (ABC)$ and $K = (AEF) \cap (ABC)$.

that $KBLC$ is harmonic, so $$-1 = (K,L;B,C) \overset{A}{=} (J,T;F,E).$$This implies $$\text{Pow}(T,(A'EF)) = ET \cdot FE = MT \cdot JT = \text{Pow}(T,(AMG)),$$so $T$ lies on the radical axis $\overline{HG}$ of $(A'EF)$ and $(AMG)$ as desired. $\blacksquare$

Redefine $T$ to be the foot of the perpendicular from $D$ to $\overline{EF}$. We will prove that it lies on the radical axis of $(A'EF)$ and $(AMG)$. Let $R$ be the second intersection of $(ABC)$ and $(AEF)$. Then it's well-known that $R,T,I,A'$ are collinear. Notice that by radical center $A'G$, $AR$, $EF$ are concurrent, say at $S$. Then $\angle AGS=180^{\circ}-\angle AGA'=90^{\circ}=\angle SMA \implies S \in (AMG)$. Also $\angle SRI-180^{\circ}-\angle ARI=90^{\circ}=\angle SMI \implies SRMI$ is cyclic. Then $\text{Pow}(T,(AMG))=ST\cdot TM=RT\cdot TI=FT \cdot TE=\text{Pow}(T,(A'EF))$, as desired.

Funny problem. Redefine $T$ to be the foot of perpendicular from $D$ to $EF$. We will prove $G,T,H$ are collinear instead, i.e. \[ \text{Pow}_T (AMG) = \text{Pow}_T (EFA') \]Apparently, $\text{Pow}_T (EFA') = TE \cdot TF$, and by letting $EF \cap (AMG) = J$, we have $\text{Pow}_T (AMG) = TM \cdot TJ$. Therefore, we need to prove \[ TE \cdot TF = TM \cdot TJ \]which is equivalent to proving $(E,F;T,J) = -1$. Let $AJ \cap (ABC) = K$. Claim 01. $J,G,A'$ collinear. Proof. Let $A'G \cap (AMG) = J'$. Since $A'$ is the antipode of $A$, we have $\measuredangle AGA' = 90^{\circ}$, and hence $\measuredangle AGJ' = 90^{\circ} = \measuredangle AMJ' = \measuredangle AMJ$, proving $J' \equiv J$. Claim 02. $K,D,Y$ collinear. Proof. By our previous claim, $J$ lies on the radical axis of $(ABC)$ and $(EFA')$, and therefore, \[ JK \cdot JA = JF \cdot JE \]which means $K = (AEF) \cap (ABC)$. Therefore, we know that $K$ is the incenter Miquel Point. Therefore, if $X$ and $Y$ are the midpoint of arcs $BC$ containing $A$ and not containing $A$ respectively, we have $K,D,Y$ collinear. By letting $AT \cap (ABC) = L$, we have $L,D,X$ by a well known lemma. Thus, \[ -1 = (X,Y;B,C) \overset{D}{=} (L,K;C,B) \overset{A}{=} (T,J;E,F) \]which is what we wanted.

Redefine $T$ to be the foot from $D$ to $EF$ and $H$ to be the second intersection of $GT$ with $(A'EF)$, and we will show $AGMH$ cyclic. Add in the point $S=(AEF) \cap (ABC)$ and let $L= AS \cap EF$. We will in fact show $G,M,H$ lie on the circle with diameter $AL$. First note $M$ lies on that circle since $AM \perp ML$ for obvious reasons. By radical axis on $(ABC),(A'EF),(AEF)$ we get $A'GL$ collinear hence $AG \perp GL$. It remains to show $AH \perp HL$. Indeed, letting $LH$ intersect $(A'EF)$ again at $K$ and noting $KH$ and $AS$ intersect on the radical axis of $(AEF)$ and $(A'EF)$, we have $ASKH$ cyclic and thus $AH \perp HK$ as desired.

Solution with hint from @above. It is well-known that $T,I,A'$ are collinear along with $(AEF)\cap (ABC)=K\ne A$. Let line $EF$ intersect $(AGM)$ again at point $J$. Observe that $AJ$ is the diameter of $(AGM)$. Moreover, since $\angle A'GA=90^\circ=\angle AGJ$, $A',G,J$ are collinear. So by radical axis theorem on $(AEF)$, $(ABC)$, $(A'EF)$, $K$ lies on $AJ$. Now $JT\cdot JM = JK\cdot JA=JE\cdot JF$, implying $(JT;EF)$ harmonic. It is well-known that $TF\cdot TE=TJ\cdot TM$ then. This implies the desired, since $T$ must lie on the radical axis of $(AHMGJ)$ and $(A'GFHE)$.

deleted as required

[asy][asy] size(8cm); 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 = -9.664267992854413, xmax = 8.910818228494016, ymin = -5.8993890210618805, ymax = 5.012432929595713; /* image dimensions */ pen qqwuqq = rgb(0,0.39215686274509803,0); pen fuqqzz = rgb(0.9568627450980393,0,0.6); pen zzttff = rgb(0.6,0.2,1); pen ffvvqq = rgb(1,0.3333333333333333,0); /* draw figures */ draw(circle((-1.7768158352132182,-0.7516822995618229), 1.7652367603555659), linewidth(0.8) + qqwuqq); draw(circle((-0.5651633926932255,-0.2764813199926565), 3.9584030633428173), linewidth(0.8) + fuqqzz); draw(circle((-2.3384079414166115,1.08415885457241), 1.919817270343408), linewidth(0.8) + qqwuqq); draw(circle((-1.2225909183064403,-2.5634401570264176), 3.1274391741754353), linewidth(0.8) + qqwuqq); draw(circle((-4.48752331183388,0.8510717143897553), 2.6078141261627823), linewidth(0.8) + qqwuqq); draw((-4.257045907673084,1.1514396538248008)--(-1.7768158352132182,-0.7516822995618229), linewidth(0.8) + linetype("4 4") + zzttff); draw((-1.8423521740436113,-2.5157020864132407)--(-2.5675883939229163,-0.1449093096982009), linewidth(0.8) + zzttff); draw((-6.075046564922025,-1.2178566162970728)--(-0.5151604411450491,0.4829376770825319), linewidth(0.8) + linetype("4 4") + ffvvqq); draw((-6.075046564922025,-1.2178566162970728)--(1.769673214613549,-3.472962639985313), linewidth(0.8) + linetype("4 4") + ffvvqq); draw((-6.075046564922025,-1.2178566162970728)--(-2.9,2.92), linewidth(0.8) + linetype("4 4") + ffvvqq); draw((-2.9,2.92)--(-3.88,-2.44), linewidth(1.2) + blue); draw((-3.88,-2.44)--(2.58,-2.68), linewidth(1.6) + blue); draw((2.58,-2.68)--(-2.9,2.92), linewidth(1.6) + blue); /* dots and labels */ dot((-2.9,2.92),dotstyle); label("$A$", (-2.8131431421552735,3.138265037307195), NE * labelscalefactor); dot((-3.88,-2.44),dotstyle); label("$B$", (-3.791875263683722,-2.234349587253223), NE * labelscalefactor); dot((2.58,-2.68),dotstyle); label("$C$", (2.6635919208656196,-2.463414551866264), NE * labelscalefactor); dot((-1.8423521740436113,-2.5157020864132407),linewidth(4pt) + dotstyle); label("$D$", (-1.7511146698584463,-2.3592941134057908), NE * labelscalefactor); dot((-0.5151604411450491,0.4829376770825319),linewidth(4pt) + dotstyle); label("$E$", (-0.4391971452564833,0.639374514255838), NE * labelscalefactor); dot((-3.513267319342443,-0.4341967670158078),linewidth(4pt) + dotstyle); label("$F$", (-3.437865772918113,-0.27688534419632627), NE * labelscalefactor); dot((-0.5651633926932255,-0.2764813199926565),linewidth(4pt) + dotstyle); label("$O$", (-0.4808453206406726,-0.11029264265956915), NE * labelscalefactor); dot((1.769673214613549,-3.472962639985313),dotstyle); label("$A'$", (1.8514525008739282,-3.2547298841658603), NE * labelscalefactor); dot((-4.241058303247535,-1.7450695599165347),linewidth(4pt) + dotstyle); label("$G$", (-4.166708842141426,-1.588802868798289), NE * labelscalefactor); dot((-2.014213880243746,0.02437045503336205),linewidth(4pt) + dotstyle); label("$M$", (-1.8135869329347303,0.2853650234902291), NE * labelscalefactor); dot((-1.7768158352132182,-0.7516822995618229),linewidth(4pt) + dotstyle); label("$I$", (-1.6886424067821624,-0.5892466595777459), NE * labelscalefactor); dot((-6.075046564922025,-1.2178566162970728),linewidth(4pt) + dotstyle); label("$K$", (-5.999228559045755,-1.047376588803828), NE * labelscalefactor); dot((-4.257045907673084,1.1514396538248008),linewidth(4pt) + dotstyle); label("$J$", (-4.166708842141426,1.3265694080949613), NE * labelscalefactor); dot((-2.5675883939229163,-0.1449093096982009),linewidth(4pt) + dotstyle); label("$T$", (-2.479957739081759,0.01465188349299872), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy][/asy] Let $EF$ meet $(AMG)$ at $K$. Notice that $$\angle AGK=\angle AMK=90^{\circ}=\angle AGA'$$hence $K,G,A'$ are collinear. Let $AK$ meet $(ABC)$ at $J$, then $$KJ\times KA=KG\times KA'=KF\times KE$$Hence $J$ lies on $(AEF)$. Redefine $T$ as the projection of $D$ on $EF$, then $$\frac{FT}{TE}=\frac{\tan\angle FDT}{\tan\angle TDE}=\frac{\tan\angle BID}{\tan\angle DIC}=\frac{BD}{DC}$$Therefore, $J$ is the center of spiral sim. sending $\overline{FTE}$ to $\overline{BDC}$. So $$\frac{JF}{JE}=\frac{FB}{EC}=\frac{BD}{DC}=\frac{FT}{TE}$$whichh implies $JT$ is the internal angle bisector of $\angle FJE$, meanwhile since $AF=AE$, $JK$ is the external angle bisector of $\angle FJE$, so $(T,H;F,E)=-1$. Therefore, $$HF\times HE=HT\times HM\hspace{20pt}(1)$$$$MT\times MH=ME^2\hspace{20pt}(2)$$We now show that $T$ lies on the radical axis of $\Omega_1=(HMG)$ and $\Omega_2=(EFA')$. Indeed, for each point $X$ on the plane define $$f(X)=Pow(X,\Omega_1)-Pow(X,\Omega_2)$$Then by linearity of PoP, $$MHf(T)=MTf(H)+HTf(M)=MT\cdot HF\cdot HE-HT\cdot ME^2=MT\cdot HT\cdot HM-HT\cdot MT\cdot MH=0$$as desired.

Great problem. I believe this works. [asy][asy] import olympiad; unitsize(45); /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(0cm); 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 = -3.4556855291888393, xmax = 7.161644883742675, ymin = -1.975717796255949, ymax = 4.150731345409496; /* image dimensions */ pen zzwwff = rgb(0.6,0.4,1); pen qqzzff = rgb(0,0.6,1); draw((0.7751464073673895,3.3086911070471117)--(0,0)--(4,0)--cycle, linewidth(0.65) + zzwwff); /* draw figures */ draw((0.7751464073673895,3.3086911070471117)--(0,0), linewidth(0.65) + zzwwff); draw((0,0)--(4,0), linewidth(0.65) + zzwwff); draw((4,0)--(0.7751464073673895,3.3086911070471117), linewidth(0.65) + zzwwff); draw(circle((2,1.2765929021365985), 2.3726966594542893), linewidth(0.65) + qqzzff); draw(circle((1.3889916156368791,1.1011928091575605), 1.1011928091575605), linewidth(0.65) + qqzzff); 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label("$B$", (0.10138532040368696,-0.1253083835583541), W * labelscalefactor); dot((4,0),dotstyle); label("$C$", (4.042977334252348,-0.11144763889395265), NE * labelscalefactor); dot((1.3889916156368791,1.1011928091575605),linewidth(4pt) + dotstyle); label("$I$", (1.4163662203482723,1.1568104978987808), NE * labelscalefactor); dot((2,1.2765929021365985),linewidth(4pt) + dotstyle); label("$O$", (2.0262389855819363,1.330069806203799), NE * labelscalefactor); dot((1.3889916156368791,0),linewidth(4pt) + dotstyle); label("$D$", (1.4424106353652614,-0.11837801122615338), E * labelscalefactor); dot((2.177579263719154,1.8697987707740351),linewidth(4pt) + dotstyle); label("$E$", (2.234150155547958,1.891429965112058), NE * labelscalefactor); dot((0.31682872142499685,1.3523746779608636),linewidth(4pt) + dotstyle); label("$F$", (0.2659244132029516,1.4478861358512114), NE * labelscalefactor); dot((3.2248535926326105,-0.7555053027739147),dotstyle); label("$A'$", (3.259845260713666,-0.8737885954360328), NE * labelscalefactor); dot((-0.14594123637954498,0.26435496183235674),linewidth(4pt) + dotstyle); label("$G$", (-0.2885053733731066,0.1380457650652736), SW * labelscalefactor); dot((1.2472039925720753,1.6110867243674494),linewidth(4pt) + dotstyle); label("$M$", (1.29854989070086,1.6835187951460362), NE * labelscalefactor); dot((1.2934839092391737,1.909888019820456),linewidth(4pt) + dotstyle); label("$H$", (1.319341007697462,1.967664060766266), NE * labelscalefactor); dot((0.9629704469345858,1.5320491096223667),linewidth(4pt) + dotstyle); label("$T$", (0.91737941242982,1.6211454441562296), NE * labelscalefactor); dot((-0.0181461969054606,2.524300947194244),linewidth(4pt) + dotstyle); label("$R$", (-0.12910680973248986,2.5498153366711276), N * labelscalefactor); dot((2,-1.0961037573176908),linewidth(4pt) + dotstyle); label("$J$", (1.9915871239209328,-1.1341679567104709), S * labelscalefactor); dot((-1.7981724538308512,0.7642504025508446),linewidth(4pt) + dotstyle); label("$K$", (-1.8379339884368798,0.6647540623125291), W * labelscalefactor); dot((1.0127252647845169,1.0614144711059215),linewidth(4pt) + dotstyle); label("$L$", (1.019389313553884,1.1338614601131567), NW * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy] Define $R$ the $A$-Sharky Devil point of $\triangle ABC.$ Let $J$ be the midpoint of $\widehat{BC}$ not containing $A,$ and $K$ is the concurrence point of radical axes on $(AFE), (GFE),$ and $(ABC).$ Note $AMGK$ is then a cyclic quadrilateral with diameter $AK$ since $\angle AMK = \angle AGK = 90^\circ.$ Extend $AT$ to meet $(AMG)$ at $L.$ By Thales Theorem $\angle ALK = 90^\circ.$ From the problem statement, $T$ lies on the radical axis of $(AMG)$ and $(A'EF).$ Therefore $TL \cdot TA = TF \cdot TE,$ and by the converse of Power of a Point, $AFLE$ is cyclic. Since $AI$ is a diameter of $(AFE)$ it follows $\angle ALI = 90^\circ,$ so $K,L,I$ are collinear. It follows $T$ is the orthocenter of $\triangle AIK.$ It follows since $IR \perp AK$ that $I,T,R$ are collinear. By the Sharky-Devil Lemma, $DT \perp EF,$ as required.

Quite nice config geo. We begin by applying radical axis to $(A'EF),(AEF),(ABC)$. Let $TI \cap (ABC)=R$, so $AR,EF,GA'$ concur at $P$. Since $\angle AGA'= \angle AMF =90$, we have that $P \in (AMG)$ (and it has diameter $AP$). We want $T\in GH$, which the radical axis of $(A'EF)$ and $(AMG)$, so we want $TF \cdot TE=TM \cdot TP \iff (P,T,F,E)=-1$. But note that $PT \cdot PM=PR \cdot PA= PF \cdot PE$, which is sufficient.

Define $T$ to be the foot of $D$ onto $EF$. We need to show that $T$ has same power w.r.t. circles $(DEF),(AMG),(A'EF)$. Define $K=MF\cap (AMG)$. Since $AM\perp EF$, we have $\angle AMK=90^\circ$, which means that $K$ is the antipode of $A$ in $(AMG)$. Since $A'$ is also the antipode of $A$ in $(ABC)$, we have $\angle AGK=\angle AGA'=90^\circ$. Hence, $A',K,G$ are collinear. Claim 1: $(K,T;E,F)=-1\Leftrightarrow T\in HG$. Proof. \[(K,T;E,F)=-1\Leftrightarrow MT\cdot MK=ME^2\Leftrightarrow TM\cdot TK=TE\cdot TF\]So, $T$ has equal power from circles $(AMG),(DEF)$. However, $T$ also has equal power from circles $(A'EF),(DEF)$ as $T\in EF$ by definition. Hence, $T$ has equal power from all three circles (as required), which means $T\in HG$. $\blacksquare$ Define $S=(AEF)\cap (ABC)$ to be the Sharkydevil Point in $ABC$. Claim 2: $K,S,A$ are collinear and $A',I,S,T$ collinear. Proof. Simply note that $K$ lies on the radical axes of circles $(AEF),(A'EF)$ and $(A'EF),(ABC)$ as established. Hence, $K$ lies on the radical axis of $(ABC),(AEF)$, which is line $AS$. So, $K\in AS$. Note that $\angle ASI=90^\circ=\angle ASA'$, which means $S,I,A'$ are collinear. Invert about the incircle to see that $(ABC)$ goes to ninepoint circle of the intouch triangle and $(AEF)$ goes to line $EF$, which means $S$ goes to $T$. So, $I,S,T$ are collinear as well. $\blacksquare$ Finally, note that taking perspective at $S$ gives $(K,T;E,F)=(A,I;E,F)=-1$, which finishes the problem by Claim 1.