Problem3

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#1 h Jul 10, 2015, 7:53 AM • 18 Y

Y by sydneymark, Davi-8191, Wizard_32, microsoft_office_word, Flow25, itslumi, somlogan, Abidabi, megarnie, HWenslawski, son7, tiendung2006, Toinfinity, Adventure10, Mango247, Rounak_iitr, Funcshun840, Mathslover2k25

Let $ABC$ be an acute triangle with $AB > AC$ . Let $\Gamma$ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$ . Let $M$ be the midpoint of $BC$ . Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$ . Assume that the points $A$ , $B$ , $C$ , $K$ and $Q$ are all different and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Proposed by Ukraine

This post has been edited 7 times. Last edited by djmathman, Feb 14, 2020, 4:21 AM
Reason: typo after 4.5 years!

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TelvCohl

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TelvCohl

#3 h Jul 10, 2015, 8:21 AM • 40 Y

Y by mssmath, anantmudgal09, AdithyaBhaskar, MariusStanean, utkarshgupta, Abubakir, CeuAzul, reveryu, DreamComeTrue, PeppaBear, Tawan, naw.ngs, samoha, Avlon, enhanced, Abbas11235, Limerent, tree_3, OlympusHero, amar_04, myh2910, PIartist, megarnie, SSaad, hakN, Assassino9931, Derpy_Creeper, Jalil_Huseynov, nguyenducmanh2705, Toinfinity, David-Vieta, OronSH, Adventure10, EpicBird08, sabkx, Sedro, Funcshun840, MS_asdfgzxcvb, Mathslover2k25, Kingsbane2139

Let $\Psi$ be the inversion with center $H$ that fixed $\odot (ABC).$ Since $Q, H, M$ and the antipode of $A$ in $\odot (ABC)$ are collinear, so $\Psi(Q)$ is the antipode of $A$ in $\odot (ABC)$ and note that $MH=M\Psi(Q)$ we get $\Psi(M)$ is the reflection of $H$ in $Q.$ Clearly, $\Psi(K)$ is the antipode of $Q$ in $\odot (ABC)$ and $\Psi(F)$ is the reflection of $H$ in $A,$ so $AQ\Psi(Q) \Psi(K)$ is a rectangle and $A\Psi(K)$ is the perpendicular bisector of $\Psi(F)\Psi(M),$ hence $\Psi(Q)\Psi(K)$ is the tangent of $\odot (\Psi(F)\Psi(K)\Psi(M))$ $\equiv$ $\Psi(\odot (FKM)).$ I.e., $\odot (HQK)$ is tangent to $\odot (FKM).$ $\qquad \blacksquare$

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oneplusone

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oneplusone

#4 h Jul 10, 2015, 9:08 AM • 50 Y

Y by buratinogigle, rkm0959, proglote, samyfinsher, BBeast, acegikmoqsuwy2000, shinichiman, toto1234567890, CatalinFetoiu, Vietnamisalwaysinmyheart, quangminhltv99, joshy0604, Garfield, utkarshgupta, baopbc, rterte, PRO2000, muffin_cowee, Ankoganit, reveryu, don2001, MathBoy2001, Lam.DL.01, Tawan, Anar24, samoha, qweDota, Adolmephus, thczarif, Supercali, sriraamster, Atpar, A-Thought-Of-God, Abbas11235, OlympusHero, myh2910, Mop2018, hakN, rayfish, Jalil_Huseynov, nguyenducmanh2705, kimyager, Adventure10, EpicBird08, Dansman2838, Gaunter_O_Dim_of_math, Math_legendno12, Sedro, Funcshun840, MS_asdfgzxcvb

Let $AE$ be the diameter of $\Gamma$ . Extend $AF$ to meet $\Gamma$ again at $D$ . Suppose $\Gamma$ has center $O$ .

Step 1: $Q,H,M,E$ are collinear.
Proof: $BECH$ is a parallelogram, so $E,M,H$ are collinear. $AE$ is the diameter, and $\angle AQE=90$ , so $Q,H,M,E$ are collinear.

Step 2: circumcircle of $\triangle DHK$ is tangent to $QE$ .
Proof: Let $QP$ be the diameter of $\Gamma$ , then $P,H,K$ are collinear. Now $\angle HKD = \angle PQD = \angle OQD = 90-\angle QBD$ . Note that $\angle QMC=\angle DMC$ , so $QBDC$ is harmonic, so $\angle QBD = \angle QMC$ . Thus $\angle HKD = 90-\angle QMC=\angle MHD$ , proving step 2.

Now let $X$ be the circumcenter of $\triangle KHD$ . Then $X$ lies on $BC$ because it is the perpendicular bisector of $HD$ , and $XH\perp QE$ . So $XK^2 = XH^2 = XF\cdot XM$ , which means $XK$ is tangent to circumcircle of $\triangle MFK$ . Also since $XK=XH$ and $XH$ is tangent to circumcircle of $\triangle QHK$ , $XK$ is also tangent to circumcircle of $\triangle QHK$ , and the result follows.

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leminscate

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leminscate

#5 h Jul 10, 2015, 10:00 AM • 6 Y

Y by samyfinsher, reveryu, Abdollahpour, e_plus_pi, Adventure10, Rayvhs

This can be done with complex numbers with $\Gamma$ as the unit circle. We construct the tangent to $(KQH)$ at $K$ and let it intersect $\Gamma$ at $X\neq K$ . Now we can prove that if $A', Q'$ are the points diametrically opposite $A,Q$ , respectively, then $A'MHQ, Q'HK$ are collinear.

$\angle XKH = \angle KQH \implies \frac{(x-k)(h-q)}{(h-k)(k-q)} \in \mathbb{R} \implies \frac{(x-k)(q+a)}{(q+k)(q-k)} \in \mathbb{R} \implies \frac{(x-k)(q+a)}{(q+k)(q-k)} = \frac{\frac{k-x}{xk}\frac{q+a}{qa}}{\frac{q+k}{qk}\frac{k-q}{qk}} \implies x=\frac{qk}{a}$ .

EDIT: Just realised that since $\angle AQK = \angle QKX$ , $QK || AX$ so $x=\frac{qk}{a}$ . So the above paragraph is unnecessary.

We want to prove that $\angle FKX=\angle FMK \iff \frac{(f-k)(k-m)}{(x-k)(b-c)} \in \mathbb{R} \iff \frac{(f-k)(k-m)}{(x-k)(b-c)} = \frac{\overline{(f-k)}\overline{(k-m)}}{\frac{k-x}{xk}\frac{c-b}{bc}} \iff (f-k)(k-m) = xkbc\overline{(f-k)}\overline{(k-m)} \iff (f-k)(k-m) = \frac{qk^2bc}{a} \overline{(f-k)}\overline{(k-m)}$ .

Now we can calculate $q=\frac{bc(2a+b+c)}{2bc+ab+ac}$ from $aq=-\frac{q-a}{\overline{(q-a)}} = \frac{h+a}{\overline{(h+a)}} = \frac{2a+b+c}{\frac{2}{a}+\frac{1}{b}+\frac{1}{c}}$ .
We calculate $k=a\frac{(a+b+c)(3bc+ab+ac)+abc}{(ab+bc+ca)(3a+b+c)+abc}$ from $kq=-\frac{k-q}{\overline{(k-q)}} = \frac{-q-h}{\overline{(-q-h)}} =q\frac{q+h}{1+q\overline{h}}$ .
The hard part is calculating $f-k$ and $k-m$ .
$k-m = \frac{(2a+b+c)[(b+c)(a^2-bc)-a(b-c)^2]}{2[(ab+bc+ca)(3a+b+c)+abc]}$ and $f-k = \frac{[(b+c)(a^2-bc)+a(b-c)^2](a+b)(a+c)}{2a[(ab+bc+ca)(3a+b+c)+abc]}$ . The main step was really finding the factorisation for the numerator of $k-m$ , which I did by somewhat guessing the factor $2a+b+c$ by inspection and luck (mostly luck). Then you can guess the form of $f-k$ so that the RTP statement holds. Here's how I sort of guessed that $2a+b+c$ was a factor:

The numerator of $k-m$ is
$\begin{align*} & 2a[(a+b+c)(3bc+ab+ac)+abc] - (b+c)[(ab+bc+ca)(3a+b+c)+abc] \\ &= 2a(a+b+c)(ab+bc+ca) + abc(6a+3b+3c) - (ab+bc+ca)(b+c)(3a+b+c) \\ &= 3abc(2a+b+c) - (ab+bc+ca)(b+c)(2a+b+c) - a(ab+bc+ca)(b+c) + 2a(a+b+c)(ab+bc+ca) \\ &= 3abc(2a+b+c) - (ab+bc+ca)(b+c)(2a+b+c) +a(ab+bc+ca)(2a+b+c). \end{align*}$ The first manipulation was done with the intention of possibly factoring out $ab+bc+ca$ , but then the $3abc(2a+b+c)$ appeared and it seemed as if the $3a+b+c$ was "pretty close" to $2a+b+c$ and that possibly $2a+b+c$ would factor out...and it did.

The remainder of the solution is basically conjugating these expressions and substituting everything in.

This post has been edited 2 times. Last edited by leminscate, Jul 11, 2015, 5:03 AM
Reason: Adding some details, fixing typos

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buratinogigle

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buratinogigle

#9 h Jul 10, 2015, 11:24 AM • 10 Y

Y by THVSH, andria, anantmudgal09, baopbc, don2001, Tawan, PIartist, Kyleray, Adventure10, Mango247

Great idea to draw circumcenter of triangle $KHD$ dear One Plus One. So we must prove that $(KHD)$ is tangent to $QE$ . I do a little different from you. We easily seen $\triangle KQH$ and $\triangle KAE$ is right and $\angle KQH=\angle KQE=\angle KAE$ , we deduce $\angle KHQ=\angle KEA=\angle KHD$ . So we are done.

I propose a general problem and solution

Let $ABC$ be an actue triangle inscribed in circle $(O)$ and $P$ is a point inside it such that $\angle BPC=180^\circ-\angle A$ . $BP,CP$ cut $CA,AB$ at $E,F$ . Circle $(AEF)$ cuts $(O)$ again at $G$ . Circle diameter $PG$ cuts $(O)$ again at $K$ . $D$ is projection of $P$ on $BC$ and $M$ is midpoint of $BC$ . Prove that the circle $(KPG)$ and circle $(KDM)$ are tangent.

Solution. Let $Q$ are symmetric of $P$ through $D$ , easily seen $Q$ lie on $(O)$ . Let $GP$ cuts $(O)$ again at $N$ . We see $\angle NPC=\angle FPG=\angle FAG=\angle BNP$ so $BN\parallel PC$ similarly, $CN\parallel BP$ . Let $AS,NR$ are diameter of $(O)$ . We eaily seen $\angle PQN=90^\circ$ so $P,Q,R$ are collinear. So $M$ is midpoint of $PN$ . We have $\angle KPG=90^\circ-\angle KGP=90^\circ-\angle KAN=\angle AKN+\angle ANK-90^\circ=\angle NKS+\angle KQA=\angle NAS+\angle KQA=\angle ANR+\angle KQA=\angle AQR+\angle KQA=\angle KQP.$

From this, $GN$ is tangent to $(KPQ)$ . Let tangent at $K,P$ of $(KPG)$ intersect at $T$ . We have $\angle KTP=180^\circ-2\angle KGP=2\angle KPG=2\angle KQP$ and $TK=TP$ , so we have $T$ is circumcenter of triangle $KPQ$ , but $BC$ is perpendicular bisector of $PQ$ , therefore $T$ lies on $BC$ . Now, $TK^2=TP^2=TD.TM$ , this means $TK$ is common tangents of $(KGP)$ and circle $(KHM)$ or they are tangent. We are done.

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THVSH

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THVSH

#12 h Jul 10, 2015, 12:16 PM • 3 Y

Y by buratinogigle, Adventure10, Mango247

Another generalization:
Let $ABC$ be a triangle with circumcircle $\odot (O)$ . A arbitrary circle $\odot (D)$ passing through $B, C$ intersects $CA, AB$ at $E, F$ , respectively. $K$ is the orthogonal projection of $D$ on $AP$ . Construct the diameter $AP$ of $\odot (AEF)$ . $\odot (AEF) \cap \odot (O)=\{A, G\}$ . $\odot (GP) \cap \odot (O)=\{G, J\}$ . Prove that $\odot (JGP)$ and $\odot (JKD)$ are tangent.

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LeVietAn

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LeVietAn

#13 h Jul 10, 2015, 12:42 PM • 2 Y

Y by Adventure10, Mango247

Another generalization

:
Let $ABC$ be a triangle with circumcircle $\Gamma$ . A arbitrary circle $(O)$ passing through $B, C$ intersects $CA, AB$ at $D, E$ , respectively. Let $H=BD\cap CE$ . Let $Q$ be the point on $\Gamma$ such that $\angle HQA =90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ=90^{\circ}$ .
Let $F$ is the orthogonal projection of $H$ on $BC$ , and let $M=KO\cap BC$ .
Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

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TelvCohl

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TelvCohl

#14 h Jul 10, 2015, 12:54 PM • 5 Y

Y by gold46, Tawan, enhanced, Adventure10, Mango247

THVSH wrote:

Another generalization:
Let $ABC$ be a triangle with circumcircle $\odot (O)$ . A arbitrary circle $\odot (D)$ passing through $B, C$ intersects $CA, AB$ at $E, F$ , respectively. $K$ is the orthogonal projection of $D$ on $AP$ . Construct the diameter $AP$ of $\odot (AEF)$ . $\odot (AEF) \cap \odot (O)=\{A, G\}$ . $\odot (GP) \cap \odot (O)=\{G, J\}$ . Prove that $\odot (JGP)$ and $\odot (JKD)$ are tangent.

The solution by oneplusone also works for this generalization

I just want to note that step 2 in the solution simplified follows from Reim theorem ( $\because AQEP$ is rectangle $\Longrightarrow QE \parallel AP$ ) .

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v_Enhance

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v_Enhance

#15 h Jul 10, 2015, 2:07 PM • 24 Y

Y by mssmath, rkm0959, silouan, Dukejukem, zschess, aayush-srivastava, huricane, don2001, Tawan, Wizard_32, naw.ngs, MathbugAOPS, Limerent, Purple_Planet, myh2910, SSaad, Assassino9931, nguyenducmanh2705, UI_MathZ_25, Adventure10, Mango247, Tellocan, Rounak_iitr, Sedro

Probably equivalent to the other inversion solutions, but anyways. . .
$[asy]size(6cm); defaultpen(fontsize(9pt)); size(8cm); pair A = dir(80); pair B = dir(220); pair C = dir(-40); draw(unitcircle, blue); pair O = circumcenter(A, B, C); pair H = orthocenter(A, B, C); pair Q = IP(CP(midpoint(A--H), H), unitcircle); pair N = midpoint(H--Q); pair T = midpoint(A--H); pair N_9 = midpoint(O--H); pair M = midpoint(B--C); pair F = foot(A, B, C); draw(circumcircle(M, F, N), darkcyan); pair L = M+T-N; pair K = OP(CP(N, H), unitcircle); draw(A--B--C--cycle, red); draw(A--F); draw(T--M--L--N, heavygreen); draw(M--Q--A, magenta); draw(Q--K--L, heavymagenta); dot("$A$", A, dir(A)); dot("$B$", B, dir(B)); dot("$C$", C, dir(C)); dot("$O$", O, dir(O)); dot("$H$", H, dir(H)); dot("$Q$", Q, dir(Q)); dot("$N$", N, dir(N)); dot("$T$", T, dir(T)); dot("$N_9$", N_9, dir(N_9)); dot("$M$", M, dir(M)); dot("$F$", F, dir(F)); dot("$L$", L, dir(L)); dot("$K$", K, dir(K)); /* Source generated by TSQ */ [/asy]$

Let $N$ , $T$ be midpoints of $HQ$ , $AH$ . Let $L$ be on the nine-point circle with $\angle HML = 90^{\circ}$ . The negative inversion at $H$ swapping $\Gamma$ and nine-point circle maps $A$ to $F$ , $K$ to $L$ , $Q$ to $M$ . AS $LM \parallel AQ$ we just need to prove $LA = LQ$ . But $MT$ is a diameter, hence $LTNM$ is a rectangle, so $LT$ passes through the center of $\Gamma$ .

This post has been edited 2 times. Last edited by v_Enhance, Jul 10, 2015, 2:39 PM
Reason: Shrink diagram, edit solution

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ISHO95

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ISHO95

#16 h Jul 10, 2015, 6:10 PM • 3 Y

Y by Mathuzb, Adventure10, Mango247

Let $R$ be the reflection of $H$ onto $BC$ and $KR$ intersects $BC$ at $T$ . We easily prove that $KQMT$ is cyclic. If $KF$ intersects circle $KQH$ at $P$ , then $Q,P,T$ are collinear. => let $l$ be line which tangents circle $KQH$ at $K$ . Then it is not hard to see line $l$ also tangents to the circle $KFM$ at $K$ . Done!

This post has been edited 2 times. Last edited by ISHO95, Jul 10, 2015, 7:56 PM

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Math-lover123

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Math-lover123

#17 h Jul 10, 2015, 7:31 PM • 2 Y

Y by mssmath, Adventure10

If $O$ is cicumcenter of $ABC$ and $V,S,T$ are midpoints of segments $OH,HK,HQ$ respectively,
then $VS$ and $VT$ are midlines of triangles $HOK$ and $HOQ$ resprectively.
So $S,T$ lie on Euler's circle of $ABC$ .
If $t$ is a line through $H$ which is perpendicular to $HM$ ,
then $t,MF,ST$ are concurrent at radical center $X$ of Euler's circle of $ABC$ and circumcircles of triangles $STH$ and $MFH$ .
Triangles $XKS$ and $XHS$ are congruent,
so $XK^2=XH^2=XS\cdot XT=XM\cdot XF$ and $\measuredangle XKS=90$ ,
which means that the line $XK$ is common tangent of circumcircles of triangles $KFM$ and $HKQ$ .

This post has been edited 2 times. Last edited by Math-lover123, Jul 10, 2015, 8:26 PM
Reason: circumcircle

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Kfp

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Kfp

#18 h Jul 10, 2015, 7:37 PM • 2 Y

Y by Adventure10, Mango247

I found this problem quite pretty.

First of all, it is well known that $Q$ , $H$ and $M$ are collinear. Assuming this, we call $H_{A}$ the second intersection of $AH$ with $\Gamma$ , and we perform an inversion of center $H$ and power $HA \cdot HH_{A}$ . After very basic angle chasing + application of the fact stated above, we get the problem transformed into the following: let $ABC$ be a triangle and let $H'$ be its incenter. Let $F'$ be its A-excenter (we know that $BF'CH'$ is cyclic for obvious reasons, we call this circle $\omega$ ), let $Q'$ be the midpoint of the circumcircle arc containing $A$ , let $M'$ be the second intersection of $Q'H$ with $\omega$ and let $K'$ be the intersection between $\Gamma^{'}$ and the perpendicular line wrt $HQ'$ passing through $Q'$ . We want to show that the circumcircle of $F'M'K'$ is tangent to $Q'K'$ at $K'$ .

To do this, we'll show that $K'$ lies on the AXIS of $F'M'$ . To show this is enough , once we defined $D$ as the center of $\omega$ , to show that $K'D$ is orthogonal to $M'F'$ . But $M'F'$ is orthogonal to $M'Q'$ ( $HF'$ being a diameter) and $M'Q'$ is orthogonal to $Q'K'$ by definition. But $Q'D'$ is a diameter of $\Gamma^{'}$ , hence done!!

This post has been edited 3 times. Last edited by Kfp, Jul 10, 2015, 9:54 PM

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mathuz

1525 posts

mathuz

#19 h Jul 10, 2015, 8:30 PM • 2 Y

Y by Davrbek, Adventure10

I think it's easy for Problem 3:

We have that two lemmas:

\textbf{Lemma 1.} The points $Q$ , $H$ and $M$ are collinear.

\textbf{Lemma 2.} Let $N$ is the midpoint of $QH$ and a point $R$ lies on the line $BC$ such that $RH\parallel AQ$ . Then $\angle RNO= 90^{\circ}$ , where $O$ is the circumcenter of the triangle $ABC$ .

We can use by two lemmas and we easily can see that the circumcircles of the triangles $KHQ$ and $KFM$ are tangent.

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proglote

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proglote

#20 h Jul 11, 2015, 5:48 AM • 2 Y

Y by Adventure10, Mango247

Here's a solution with inversion WRT $Q$ . First, let $A_0$ denote the antipode of $A$ in $\Gamma$ ; we know that $Q, H, M, A_0$ are collinear and $M$ is the midpoint of $HA_0$ , so that $M'$ becomes the harmonic conjugate of $Q$ WRT $A'_0, H'$ (Note that $A', M', F'$ are collinear since $(QAMF)$ is cyclic):
$[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 = -4.800874036810980, xmax = 34.89250840395043, ymin = -7.788624316942659, ymax = 11.96084728234451; /* image dimensions */ pen qqttqq = rgb(0.000000000000000,0.2000000000000002,0.000000000000000); pen aqaqaq = rgb(0.6274509803921575,0.6274509803921575,0.6274509803921575); pen qqwuqq = rgb(0.000000000000000,0.3921568627450985,0.000000000000000); draw((558.4074274619950,-167.6931808709421)--(558.4848138116372,-168.2773169816810)--(559.0689499223762,-168.1999306320388)--(558.9915635727340,-167.6157945212999)--cycle, qqwuqq); draw((11.94814897609412,-2.701786992093843)--(11.83986216803928,-3.280991282635144)--(12.41906645858058,-3.389278090689988)--(12.52735326663543,-2.810073800148687)--cycle, qqwuqq); draw((12.24038502123578,-5.722950743110142)--(11.76393233897595,-5.376257096575377)--(11.41723869244118,-5.852709778835205)--(11.89369137470101,-6.199403425369972)--cycle, qqwuqq); draw((18.65735096727185,3.095730151442312)--(19.13380364953167,2.749036504907546)--(19.48049729606644,3.225489187167374)--(19.00404461380661,3.572182833702140)--cycle, qqwuqq); /* draw figures */ draw(circle((11.92149259612108,3.614427735914360), 7.082678004468662), linewidth(1.200000000000002) + qqttqq); draw((11.89369137470101,-6.199403425369972)--(16.80965888055248,-1.511014497759957), aqaqaq); draw((7.004365667404278,-1.483237463161803)--(11.89369137470101,-6.199403425369972), aqaqaq); draw((19.00404461380661,3.572182833702140)--(11.89369137470101,-6.199403425369972)); draw((11.53043691294743,-3.457446355699668)--(11.89369137470101,-6.199403425369972)); draw((5.406156668775245,-1.478709958916395)--(11.89369137470101,-6.199403425369972)); draw((5.406156668775245,-1.478709958916395)--(26.02563863289654,-1.537122089126370)); draw((19.00404461380661,3.572182833702140)--(26.02563863289654,-1.537122089126370)); draw((26.02563863289654,-1.537122089126370)--(8.649924021738130,-3.839056692143185)); draw((5.406156668775245,-1.478709958916395)--(14.14050284768400,-3.111664830980842)); draw((11.89369137470101,-6.199403425369972)--(12.52735326663543,-2.810073800148687)); /* dots and labels */ dot((19.00404461380661,3.572182833702140),dotstyle); label("$Q$", (19.11515275219922,3.738845041291063), NE * labelscalefactor); dot((7.004365667404278,-1.483237463161803),dotstyle); label("$C'$", (7.143250840395042,-1.205467117180268), NE * labelscalefactor); dot((16.80965888055248,-1.511014497759957),dotstyle); label("$B'$", (16.42078039617833,-1.122136013385808), NE * labelscalefactor); dot((11.89369137470101,-6.199403425369972),dotstyle); label("$H'$", (12.25422520645530,-6.177556310249753), NE * labelscalefactor); dot((26.02563863289654,-1.537122089126370),dotstyle); label("$A'$", (26.14274250553207,-1.372129324769190), NE * labelscalefactor); dot((15.30832042658575,-1.506761414320957),dotstyle); label("$A'_0$", (14.55971907810204,-1.177690082582115), NE * labelscalefactor); dot((14.14050284768400,-3.111664830980842),dotstyle); label("$M'$", (14.19861762832604,-3.455406919630706), NE * labelscalefactor); dot((11.53043691294743,-3.457446355699668),dotstyle); label("$F'$", (10.92092754574392,-3.260967677443630), NE * labelscalefactor); dot((5.406156668775245,-1.478709958916395),dotstyle); label("$K'$", (5.504405799103985,-1.316575255572882), NE * labelscalefactor); dot((8.649924021738130,-3.839056692143185),dotstyle); label("$M_0$", (8.143224085928571,-4.233163888379004), SW * 0.1*labelscalefactor); dot((12.52735326663543,-2.810073800148687),dotstyle); label("$X$", (12.64310369082944,-2.649872916284253), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]$
Note that $A'(Q, M'; A'_0, H')$ is a harmonic pencil and $A'Q \parallel H'K' \Longrightarrow A'M'$ cuts $H'K'$ at its midpoint $M_0$ . Define $X$ as the foot of the perpendicular from $H'$ to $M'K'$ , so that $(H'M'XF')$ all lie on the circle with diameter $H'M'$ . Now $\angle K' XF' = \angle F'H'M' = 180 - \angle K'M_0 F'$ , i.e. $(K' X F' M_0)$ is cyclic. But note that $M_0 X^2 = M_0 H'^2 = M_0 F' \cdot M_0 M'$ , i.e. $M_0 X$ is tangent to $(M' X F')$ and $\angle H' K' F' = \angle M_0 X F' = \angle K' M' F'$ , i.e. $H'K'$ is tangent to $(F'K'M')$ , i.e. $(KQH)$ is tangent to $(FKM)$ at $K$ , as desired.

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Géza Kós

111 posts

Géza Kós

#24 h Jul 11, 2015, 11:01 AM • 2 Y

Y by Adventure10, Mango247

I can be found quickly that $Q$ is the intersection of $\Gamma$ and the ray $MH$ .

What happens if, by accident, we take the other intersection point? Instead of $Q$ and $K$ , we will get the reflections of $H$ about $M$ and $F$ , respectively and find that the two circles in the problem statement are trivially tangent to each other.

Now let us draw both sides of the problem in the same picture. We can realize what object can connect the two parts...

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Shreyasharma

688 posts

Shreyasharma

#135 h Sep 10, 2024, 1:40 AM

Y by

20 minute solve, took time to find the right inversion.

Note that $Q$ , $H$ and $M$ are collinear from reflecting $H$ about $M$ to $H'$ and noting that $AH'$ is a diameter of $(ABC)$ . Now consider the inversion with power $AH \cdot HF$ , which swaps the following pairs of points:

The pairs $(A, F)$ , $(B, E)$ and $(C, D)$ where $D$ and $E$ are the feet from $B$ and $C$ to $AC$ and $AB$ respectively.
The pair $(Q, M)$ , as $\angle HQA = \angle HFM = 90$ .
The pair $(K, N)$ , where $N$ denotes the point on $(DEF)$ such that $\angle HMN = 90$ .

Hence we wish to show $MN$ is tangent to $(NAQ)$ - however noting that $MN \parallel AQ$ as they are both perpendicular to $QM$ , it follows that it suffices to show $NA = NQ$ . This would follow immediately if we could show $2MN = AQ$ , yet this is clear from the homothety at $H$ with scale factor $2$ sending $(DEF)$ to $(ABC)$ , finishing. $\square$

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Mathandski

777 posts

Mathandski

#136 h Sep 27, 2024, 4:38 PM

Y by

I inverted at H

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SomeonesPenguin

129 posts

SomeonesPenguin

#137 h Oct 31, 2024, 10:09 PM • 1 Y

Y by zzSpartan

So this took about 10 minutes.

Let $O$ be the circumcenter, $A'$ be the $A$ antipode, $N$ be the midpoint of $AH$ and $T$ lies on the nine-point circle such that $TM\perp HM$ .

It is known that $\overline{A'-M-H-Q}$ . Now invert at $H$ with radius $\sqrt{-HA\cdot HF}$ . It's easy to see that $Q\mapsto M$ , $K\mapsto T$ and $A\mapsto F$ .

After inversion, it suffices to prove that $MT$ is tangent to $(AQT)$ . From middle lines, $ON$ is parallel to $A'Q$ , hence $ON\perp AQ$ so $ON$ is the perpendicular bisector of $AQ$ . Now notice that $NM$ is a diameter in the nine-point circle so $\angle NTM=\angle TMH=90^\circ$ , or $NT\parallel MH$ . Therefore, we get $\overline{T-O-N}$ and since $ON$ is the perpendicular bisector of $AQ$ , we get the desired conclusion. $\blacksquare$

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ihatemath123

3496 posts

ihatemath123

#138 h Nov 7, 2024, 5:07 AM • 1 Y

Y by OronSH

Invert at $H$ .

Let $N^*$ be the minor arc midpoint of arc $B^{*}C^*$ in $(A^*B^*C^*)$ . Since $N^*$ and $Q^*$ are antipodal arc midpoints in $(A^*B^*C^*)$ , it follows that $\angle N^* K^* Q^* = 90^{\circ}$ . We also have $\angle K^* Q^* H = 90^{\circ}$ by definition. It's well known that the uninverted $M$ , $H$ and $Q$ are collinear, so $M^*$ , $H$ and $Q^*$ are collinear – since $M^*$ lies on $(HB^*C^*)$ , it follows that $M^*$ is the foot from $F^*$ to line $HQ^*$ .

Now, what we want to show is clearly true: $\overline{Q^*K^*} \parallel \overline{M^* F^*}$ and $M^* K^* = F^* K^*$ since line $N^* K^*$ is the perpendicular bisector of $\overline{M^* F^*}$ . Therefore, $(F^* K^* M^* )$ is tangent to line $KQ$ .

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iStud

272 posts

iStud

#140 h Jan 30, 2025, 8:24 AM

Y by

reim's spam lol

Note that it suffices to prove that $\angle{MFK}-\angle{QHK}=\angle{MKQ}=90^\circ+\angle{HKM}$ .

Let $O$ be the circumcenter of $\triangle{ABC}$ . Define $A',Q'$ to be the antipodes of $A,Q$ , respectively .Notice that $Q$ is the $A$ -queue point of $\triangle{ABC}$ , so it's well known that $\overline{Q,H,M,A'}$ . Moreover, $\overline{K,H,Q'}$ . Let $J$ be the midpoint of $Q'H$ . Note that $AF\cap(ABC)$ is the reflection of $H$ over $BC$ and $A'$ is the reflection of $H$ over $M$ . Using Reim's Theorem 2 times, we conclude that $AJFK$ and $MJQK$ are cyclic.

Let $\angle{HQK}=\alpha$ and $\angle{JQM}=\beta$ . Note that $J$ must be lying on the mid parallel line of $AQ$ and $A'Q'$ (which passes through $O$ ), so $\angle{AJQ}=2\angle{JQM}=2\beta$ . We have $JO\parallel HQ$ , so $\angle{AFK}=\angle{AJK}=\angle{AJO}+\angle{OJK}=\frac{\angle{AJQ}}{2}+\angle{QHK}=90^\circ-\alpha-\beta$ , then $\angle{MFK}=90^\circ+\angle{AFK}=180^\circ-\alpha+\beta$ . Lastly, using information that $\angle{QHK}=90^\circ-\alpha$ , we can have $\angle{MFK}-\angle{QHK}=90+\beta=90^\circ+\angle{JQM}=90^\circ+\angle{JKM}=90^\circ+\angle{HKM}$ , as desired. $\blacksquare$

P2 of GOWACO 2021 was definitely much harder than this one

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Ilikeminecraft

738 posts

Ilikeminecraft

#141 h Feb 22, 2025, 4:25 AM

Y by

rename appropriately first( $F \to D, Q\to D$ ).
Invert about $H$ first with radius $-\sqrt{AH\cdot HD}.$ Then, the circumcircle maps onto the 9-point circle, $K$ maps to intersection of 9 point circle and the line perpendicular to $GM$ that passes through $M.$ Clearly, $G\leftrightarrow M$ . Problem simplifies to proving that $(AGK^*)$ is tangent to $MK^*.$
Then, reflect $H$ across $K^*$ and $M,$ and they both land on the circumcircle. In fact, they form a rectangle. This finishes.

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cj13609517288

1940 posts

cj13609517288

#142 h Feb 25, 2025, 4:14 PM • 1 Y

Y by MS_asdfgzxcvb

This works????????

Let $\ell$ be the line through $M$ perpendicular to $HM$ , and let $S$ be the point on $\ell$ such that $(QAS)$ is tangent to $\ell$ . Since $QA\perp QM$ , $S$ is the projection of the midpoint of $QA$ onto $\ell$ . So in complex coordinates with $(ABC)$ as the unit circle,
$s=m+\frac{a-q}{2}\Longrightarrow 2s-h=2m+a-q-h=-q.$ Therefore, the reflection of $H$ over $S$ is exactly the $Q$ -antipode, meaning that $K$ lies on $HS$ .

Now consider the negative inversion centered at $H$ swapping $A$ and $F$ . Then it also swaps $Q$ and $M$ , and $K$ and $S$ because they are collinear with $H$ and also $\ell$ swaps with $(QH)$ . So since $(QAS)$ is tangent to $\ell$ , we get $(MFK)$ is tangent to $(QH)$ . $\blacksquare$

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VideoCake

19 posts

VideoCake

#143 h Apr 7, 2025, 12:50 PM

Y by

Solution. Let $A'$ be the antipode of $A$ on $\Gamma$ . Observe that $H, M, A'$ are collinear (as $M$ is known to be the midpoint of $HA'$ ) and observe that $Q, H, A'$ are collinear (as $AQ \perp QH$ ). Thus, $Q, M, H$ are collinear. Let $D, E$ be the foots of the altitudes from $B, C$ .
Perform an inversion around a circle with center $H$ . Denote $X^*$ to be the inverted image of $X$ . We get that

$F^*D^*E^*$ is a triangle with orthic triangle $A^*B^*C^*$ and orthocenter $H^*$ .
$Q^*$ lies on the line $A^*D^*$ and on the circle $(A^*B^*C^*)$ , as $AQDH$ is cyclic.
$K^*$ is on the perpendicular line to $H^*Q^*$ through $Q^*$ and on $(A^*B^*C^*)$ , as $K$ lies on the circle with diameter $QH$ .
$M^*$ is the intersection of $(H^*B^*C^*F^*)$ and $Q^*M^*$ , as line $BC$ goes to the circle $(H^*B^*C^*)$ .

It is enough to show that $(M^*F^*K^*)$ is tangent to $Q^*K^*$ . For simplicity, we remove the $*$ symbols from the points and solve the remaining problem. Let $O$ be the circumcenter of $(BMFCH)$ .
First, it is known that $H$ is the incenter of $\Delta ABC$ , and that $D$ , $E$ and $F$ are the three excenters of $\Delta ABC$ . This means that $O$ is the $A$ -southpole and lies on the circle $(ABC)$ . In particular, $O$ is the midpoint of $HF$ . As $FM \perp MH$ , we have $OMF$ isosceles. As $AQ$ is the exterior angle bisector of $\angle BAC$ , we know that $Q$ is the northpole. Therefore, $OQ$ is the diameter of $ABC$ , meaning $QK \perp KO$ . Notice that $KO$ is also the perpendicular bisector of $MF$ , so $KMF$ is isosceles. Notice that $FM \perp MQ \perp QK \implies MF \parallel QK$ . Thus, $\angle QKM = \angle FMK = \angle KFM$ , and $(MFK)$ is indeed tangent to $QK$ , as desired.

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Retemoeg

64 posts

Retemoeg

#144 h Apr 20, 2025, 5:17 AM

Y by

Let $AF$ intersect $(O)$ twice at $H'$ . Denote $L$ the orthogonal projection of $Q$ onto $AD$ . We'll call the midpoints of segments $QL, QA, QH$ , $Y, X, I$ resp. Let $Z$ be the center of $(KDM)$ . Denote $A'$ the antipodal point of $A$ in $(O)$ .

We readily notice that $\angle AQH = \angle AQA' = 90^{\circ}$ , thus $Q, H, A'$ are collinear. From a common result: $H, M, A'$ are collinear. Hence all these four points lies on a straight line. We'll show that $\angle MKI = \angle ZKM$ , i.e $\angle MKI = \angle KFA = \angle H'KF + \angle KH'F$ . A spiral similarity $\phi$ centered at $K$ sends $Q$ to $A$ and $H$ to $A'$ , thus:
$\angle MKI = \angle MKH + \angle HKI = \angle MKH + \angle QHK = \angle MKH + \angle AA'K = \angle MKH + \angle AH'K$ So we'll aim to show that $\angle MKH = \angle H'KF$ .
Observe that $\phi$ also sends $F$ to $Y$ , $H'$ to $L$ , $M$ to $X$ . Thus, $\angle MKH = \angle XKQ$ and $\angle H'KF = \angle LKY$ . And to finish off the problem, $X$ is the intersection of the two tangents from $Q$ and $L$ w.r.t $(I)$ . Thus $KX$ is the $K$ -symmedian w.r.t $\triangle QKL$ , hence $\angle QKX = \angle LKY$ and we are done.

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bjump

1070 posts

bjump

#145 h May 5, 2025, 4:27 PM

Y by

Consider a Negative inversion at $H$ such that the circumcircle of $\triangle ABC$ is mapped to the Nine Point Circle of $\triangle ABC$ . $F$ is swapped with $A$ under this inversion. $Q$ is mapped to $M$ and vice versa. $K$ is mapped to the point on $(A'B'C')$ such that $\angle HQ'K' = 90^\circ$ . Since $M' \in (HB'C')$ we have $\angle F'M'H = \angle F'M'Q' = \angle M'Q'K' = 90^\circ$ , so $Q'K' \parallel M'F'$ , Note that the $Q'$ -antipode in $(A'B'C')$ is the midpoint of $HF'$ , $K'$ lies on the perpendicular bisector of $F'M'$ therefore $Q'K'$ is tangent to $(F'M'K')$ . Uninverting gives the desired conclusion.

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fearsum_fyz

91 posts

fearsum_fyz

#146 h May 17, 2025, 12:02 PM

Y by

Consider the composition $\mathcal{T}$ of inversion centered at $H$ with radius $\sqrt{HA \cdot HF}$ and reflection over $H$ .
It is easy to see that $\mathcal{T}$ swaps the circumcircle and nine point circle, sending $A$ to $F$ and $Q$ to $M$ .
Let $K^*$ be the image of $K$ under $\mathcal{T}$ .
Then $\measuredangle{QMK^*} = \measuredangle{HMK^*} = \measuredangle{HKQ} = 90^{\circ}$ , so $AQ \parallel MK^*$ . Hence it would suffice to show that $A K^* = Q K^*$ .
Let $Q'$ be the antipode of $Q$ . By homothety at $H$ , $K^*$ is the midpoint of $HQ'$ .
By Apollonius's theorem in $\Delta{AQ'H}$ and $\Delta{QQ'H}$ , we have
$2 A{K^*}^2 + 2 H{K^*}^2 = AQ'^2 + AH^2$
$2 Q{K^*}^2 + 2 H{K^*}^2 = QQ'^2 + QH^2$
Subtracting and using the Pythagorean theorem, we get
$2 A{K^*}^2 - 2 Q{K^*}^2 = AQ^2 - AQ^2 = 0$
as desired.

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zuat.e

96 posts

zuat.e

#147 h May 20, 2025, 9:04 PM • 1 Y

Y by Kingsbane2139

Consider the inversion at $H$ with radius $r=\sqrt{AH\cdot AF}$ followed by a reflection w.r.t $H$ .

It suffices to show that $K'M$ is tangent to $(AQK')$ , where $K'$ is the inverse of $K$ .

Let $V,W$ be the reflections of $H$ across $K',M$ , so as $AQWV$ is cyclic (because $VH\cdot HK=2K'H\cdot HK=2HQ\cdot HM=HQ\cdot HW$ and $W\in (ABC)$ ) and $\measuredangle AQW=\measuredangle QWV=90º$ , we have $AQWV$ rectangle, yet clearly $K'$ is center of $\triangle VHW$ , from where it is clear that $\triangle K'AQ$ is isosceles, consequently $\measuredangle MK'Q=\measuredangle K'QA=\measuredangle K'AQ$ , as desired.

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cursed_tangent1434

737 posts

cursed_tangent1434

#148 h Jul 10, 2025, 4:54 AM

Y by

As a true config main we refrain from indulging in inversion. We start off by noting the following which allows us to adjust our frame of reference.

Claim : Point $H$ is the $Q-$ Humpty Point in $\triangle QBC$ .

Proof : Simply note that if the $A-$ antipode is $A'$ ,
$\measuredangle HBC = \measuredangle CAF = \measuredangle A'AB = \measuredangle A'QB = \measuredangle HQB$ a similar argument shows that $\measuredangle BCH = \measuredangle CQH$ which suffices to show the claim.

Now, we rewrite the problem with $\triangle QBC$ as our reference triangle.

Reference Shift wrote:

Let $\triangle ABC$ be a triangle with $A-$ Humpty point $H_a$ . Let $M$ denote the midpoint of segment $BC$ and $F$ the foot of the altitude from $H_a$ to $BC$ . Let $K$ denote the second intersection of $(AH_a)$ and $(ABC)$ . Show that circles $(AH_aK)$ and $(KFM)$ are tangent.

The following is the key claim.

Claim : Lines $BK$ and $CH_a$ intersect on $(AH_a)$ (and similarly).

Proof : We show the proof for one as the other is entirely similar. Note,
$\measuredangle KBC + \measuredangle BCH_a = \measuredangle KAC + \measuredangle CAH_a = \measuredangle KAH_a$ which implies that $BK \cap CH_a$ lies on $(AH_a)$ as desired.

Let $X= BK \cap CH_a$ and $Y = CK \cap BH_a$ . Now, by Pascal's Theorem on concyclic hexagon $H_aH_aYKKX$ it follows that the tangents to $(AH_a)$ at $K$ and $H_a$ intersect on $\overline{BC}$ .

The tangent to $(AH_a)$ at $H_a$ is perpendicular to $AM$ and hence must pass through $H$ . Thus, this intersection of tangents is the $A-$ Ex point of $\triangle ABC$ . Thus, if the tangent intersection point is $T$ ,
$TK^2 = TH_a^2 = TF \cdot TM$ which implies that $\overline{TK}$ is also tangent to $(KFM)$ and the result follows.

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ihategeo_1969

285 posts

ihategeo_1969

#149 h Jul 11, 2025, 12:12 AM

Y by

Changing labels.

Quote:

Let $\triangle ABC$ have orthic triangle $\triangle DEF$ , orthocenter $H$ , midpoint of $\overline{BC}$ as $M$ . Let $Q_A$ be its $A$ -Queue point. Let $K=(Q_AH) \cap (ABC)$ . If $AB<AC$ and $A,Q_A,K,B,C$ are cyclic in this order; then prove that $(KQ_AH)$ and $(DKM)$ are tangent to each other.

Invert about $H$ swapping $(ABC)$ and $(DEF)$ which means $K^*=\overline{M\infty_{\perp HM}} \cap (DEF)$ . Let $N$ be midpoint of $\overline{AH}$ . Now $\measuredangle NK^*M=\measuredangle NDM=90^\circ$ ; which means $\overline{NK^*} \parallel \overline{HM} \perp \overline{AQ_A} \parallel \overline{MK^*}$ which means $K^*A=K^*Q_A$ . This implies $\overline{MK^*}$ is tangent to $(AQ_AK^*)$ and invert back and done.