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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 2025
0 replies
Show that three lines concur
benjaminchew13   2
N a minute ago by benjaminchew13
Source: Revenge JOM 2025 P2
t $A B C$ be a triangle. $M$ is the midpoint of segment $B C$, and points $E$, $F$ are selected on sides $A B$, $A C$ respectively such that $E$, $F$, $M$ are collinear. The circumcircles $(A B C)$ and $(A E F)$ intersect at a point $P != A$. The circumcircle $(A P M)$ intersects line $B C$ again at a point $D != M$. Show that the lines $A D$, $E F$ and the tangent to $(A E F)$ at point $P$ concur.
2 replies
benjaminchew13
13 minutes ago
benjaminchew13
a minute ago
slightly easy NT fe
benjaminchew13   2
N 2 minutes ago by benjaminchew13
Source: Revenge JOM 2025 P1
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$f(a) + f(b) + f(c) | a^2 + af(b) + cf(a)$$for all $a, b, c\in\mathbb{N}$
2 replies
benjaminchew13
16 minutes ago
benjaminchew13
2 minutes ago
Cheesy's math casino
benjaminchew13   1
N 4 minutes ago by benjaminchew13
Source: Revenge JOM 2025 P4
There are $p$ people playing a game at Cheesy's math casino, where $p$ is an odd prime number. Let $n$ be a positive integer. A subset of length $s$ from the set of integers from $1$ to $n$ inclusive is randomly chosen, with an equal probability ($s <= n$ and is fixed). The winner of Cheesy's game is person $i$, if the sum of the chosen numbers are congruent to $i mod p$ for $0 <= i <= p - 1$.

For each $n$, find all values of $s$ such that no one will sue Cheesy for creating unfair games (i.e. all the winning outcomes are equally likely).
1 reply
1 viewing
benjaminchew13
10 minutes ago
benjaminchew13
4 minutes ago
2013 Japan MO Finals
parkjungmin   0
4 minutes ago
help me

we cad do it
0 replies
parkjungmin
4 minutes ago
0 replies
Square number
linkxink0603   4
N 4 hours ago by pooh123
Find m is positive interger such that m^4+3^m is square number
4 replies
linkxink0603
Yesterday at 11:20 AM
pooh123
4 hours ago
Inequalities
sqing   7
N 5 hours ago by sqing
Let $ a,b>0, a^2+ab+b^2 \geq 6  $. Prove that
$$a^4+ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10}  $. Prove that
$$a^4+ab+b^4  \leq 10$$Let $ a,b>0,  a^2+ab+b^2 \geq \frac{15}{2}  $. Prove that
$$ a^4-ab+b^4\geq 10$$Let $ a,b>0,  a^2+ab+b^2 \leq \sqrt{10}  $. Prove that
$$-\frac{1}{8}\leq  a^4-ab+b^4\leq 10$$
7 replies
sqing
Thursday at 2:42 PM
sqing
5 hours ago
Compilation of functions problems
Saucepan_man02   2
N Today at 12:45 AM by Saucepan_man02
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..
2 replies
Saucepan_man02
May 7, 2025
Saucepan_man02
Today at 12:45 AM
How many triangles
Ecrin_eren   5
N Today at 12:10 AM by jasperE3


"Inside a triangle, 2025 points are placed, and each point is connected to the vertices of the smallest triangle that contains it. In the final state, how many small triangles are formed?"


5 replies
Ecrin_eren
May 2, 2025
jasperE3
Today at 12:10 AM
Triangle on a tetrahedron
vanstraelen   2
N Yesterday at 7:51 PM by ReticulatedPython

Given a regular tetrahedron $(A,BCD)$ with edges $l$.
Construct at the apex $A$ three perpendiculars to the three lateral faces.
Take a point on each perpendicular at a distance $l$ from the apex such that these three points lie above the apex.
Calculate the lenghts of the sides of the triangle.
2 replies
vanstraelen
Yesterday at 2:43 PM
ReticulatedPython
Yesterday at 7:51 PM
shadow of a cylinder, shadow of a cone
vanstraelen   2
N Yesterday at 6:33 PM by vanstraelen

a) Given is a right cylinder of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the botom base?

b) Given is a right cone of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the base?
2 replies
vanstraelen
Yesterday at 3:08 PM
vanstraelen
Yesterday at 6:33 PM
2023 Official Mock NAIME #15 f(f(f(x))) = f(f(x))
parmenides51   3
N Yesterday at 5:13 PM by jasperE3
How many non-bijective functions $f$ exist that satisfy $f(f(f(x))) = f(f(x))$ for all real $x$ and the domain of f is strictly within the set of $\{1,2,3,5,6,7,9\}$, the range being $\{1,2,4,6,7,8,9\}$?

Even though this is an AIME problem, a proof is mandatory for full credit. Constants must be ignored as we dont want an infinite number of solutions.
3 replies
parmenides51
Dec 4, 2023
jasperE3
Yesterday at 5:13 PM
Geometry
AlexCenteno2007   3
N Yesterday at 4:18 PM by AlexCenteno2007
Let ABC be an acute triangle and let D, E and F be the feet of the altitudes from A, B and C respectively. The straight line EF and the circumcircle of ABC intersect at P such that F is between E and P, the straight lines BP and DF intersect at Q. Show that if ED = EP then CQ and DP are parallel.
3 replies
AlexCenteno2007
Apr 28, 2025
AlexCenteno2007
Yesterday at 4:18 PM
Cube Sphere
vanstraelen   4
N Yesterday at 2:37 PM by pieMax2713

Given the cube $\left(\begin{array}{ll} EFGH \\ ABCD \end{array}\right)$ with edge $6$ cm.
Find the volume of the sphere passing through $A,B,C,D$ and tangent to the plane $(EFGH)$.
4 replies
vanstraelen
Yesterday at 1:10 PM
pieMax2713
Yesterday at 2:37 PM
Combinatorics
AlexCenteno2007   0
Yesterday at 2:05 PM
Adrian and Bertrand take turns as follows: Adrian starts with a pile of ($n\geq 3$) stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.
0 replies
AlexCenteno2007
Yesterday at 2:05 PM
0 replies
Concurrence, Isogonality
Wictro   40
N Apr 21, 2025 by CatinoBarbaraCombinatoric
Source: BMO 2019, Problem 3
Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that:
$1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively.
$2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively.
Prove that $KL$ and $ST$ intersect on the line $BC$.
40 replies
Wictro
May 2, 2019
CatinoBarbaraCombinatoric
Apr 21, 2025
Concurrence, Isogonality
G H J
G H BBookmark kLocked kLocked NReply
Source: BMO 2019, Problem 3
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Wictro
119 posts
#1 • 5 Y
Y by Circumcircle, albgeo, vinhhop, trying_to_solve_br, Adventure10
Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that:
$1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively.
$2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively.
Prove that $KL$ and $ST$ intersect on the line $BC$.
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fastlikearabbit
28616 posts
#2 • 7 Y
Y by Snakes, Valikk202, nhusanboev, mNT, a_friendwr_a, Adventure10, Stuffybear
Invert through $A$ with radius $\sqrt{AB\cdot AC}$ and then reflect everything across the bisector of angle $\angle{BAC}$. Denote this transformation by $\Phi$. Let $\Phi(P)=P’$ for every point $P$ in the plane. Let $D$ be the antipodal of $ A$ in $\odot(ABC)$. Points $K’, T’, S’, L’$ are just the intersection points of $AX$ and $AY$ with $BD$ and $CD.$ Observe that $\angle{DS’L’}=\angle{BS’A}=\angle{AT’C}$ because $\triangle{ABS’} \sim \triangle{ACT’}$. The last equality of angles implies that $K’T’L’S’$ is cyclic. We need to prove that $ \odot(ABC)$, $\odot(AS’T’)$, and $\odot(AK’L’)$ have a common point, different from $A$. The last is equivalent with proving that $$ \dfrac{pow_{\odot(ABC)} (K’)}{pow_{\odot(ABC)}(L’)}=\dfrac{pow_{\odot(AS’T’)}(K’)}{pow_{\odot(AS’T’)}(L’)}.$$Notice that $\dfrac{pow_{\odot(ABC)} (K’)}{pow_{\odot(ABC)}(L’)}=\dfrac{K’A \cdot K’X’}{L’A\cdot L’Y’}$, and $\dfrac{pow_{\odot(AS’T’)}(K’)}{pow_{\odot(AS’T’)}(L’)}=\dfrac{K’A\cdot K’T’}{L’A \cdot L’S’}.$ So, we have to prove that $\dfrac{K’T’}{K’X’}=\dfrac{L’S’}{L’Y’} \iff \dfrac{X’T’}{X’K’}=\dfrac{Y’S’}{Y’L’}$, or, in other words, that $X’$ and $Y’$ are corresponding points in the similar triangles $\triangle{DK’T’}$ and $\triangle{DS’L’}$, respectively. The problem is solved as $X’$ is the foot of the perpendicular from $D$ to $S’L’$, and $Y’$ is the foot of the perpendicular from $D$ to $K’T’$.
This post has been edited 3 times. Last edited by fastlikearabbit, May 6, 2019, 12:51 PM
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Optx
4 posts
#3 • 7 Y
Y by MathsLion, Intolek, HECAM-CA-CEBEPA, a_friendwr_a, Adventure10, Mango247, Stuffybear
Invert in A with any radius. Consider cyclic complete quadrilateral KTSL with extensions of opposite sides. Let Q be Miquel point of this complete quadrilateral. From well known lemma it lies on AM where M is other intersection of opposite sides of KTSL. Let R be intersection of diagonals of KTSL. From another lemma O,R and Q are collinear where O is center of KTSL. Also OR is perpendicular to AM so Q is on circle with diameter AR. B and C are obviously also on this circle so we are done.
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Pluto1708
1107 posts
#4 • 2 Y
Y by a_friendwr_a, Adventure10
Note that $\odot{AKL},\odot{AST}$ are concyclic.We will show $\odot{KLST}$ cyclic and then the result follows from radical axis theorm.$AS\cdot AL=AK\cdot AT 	\Longleftrightarrow AB.AC.\cos CAY \cdot \cos BAY=AB.AC \cos BAX \cos CAX$ which is obviously true.$\blacksquare$
EDIT-This doesn't complete it I'll post the correct solution later
This post has been edited 1 time. Last edited by Pluto1708, May 3, 2019, 6:15 AM
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Flash_Sloth
230 posts
#7 • 10 Y
Y by AlastorMoody, HECAM-CA-CEBEPA, HolyMath, BatyrKHAN, a_friendwr_a, Adventure10, Mango247, Gaunter_O_Dim_of_math, Stuffybear, NuMBeRaToRiC
Let $H$ be the feet of $A$ on $BC$. Then $A, L, T, H, C$ concyclic and $A, B, S, K, T$ concyclic. It is clear from the assumption that $\triangle ATC \sim \triangle ASB$ and $\triangle AKB \sim \triangle ALC$, which implies $AL \cdot AS = AK \cdot AT$, thus $L, S, K, T$ is concyclic. Furthermore, let $M$ be the midpoint of BC, then by simple reflexion argument $ML =MS$ and $MT =MK$, which means $M$ is the center of $\odot (LSKT)$.

Let $P= ST \cap LK$, $Q = LT \cap SK$. By radical axis theorem, $AH$, $LT$, $SK$ are concurrent , thus $Q$ lies on $AH$, which is perpendicular to BC. Moreover, by Brocard's thm, $MP \perp AQ$, therefore $P$ lies on $BC$ since $M$ lies on $BC$.
Attachments:
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RC.
439 posts
#8 • 3 Y
Y by parola, a_friendwr_a, Adventure10
No body tempted to overkill this thing yet?
Let \(BK \cap CL = M, BS\cap CT= N, LT \cap KS = O\).
By Desargues' on \(\Delta BKS\) and \(\Delta CLT\), it suffices to prove that \(\overline{M,O,N}.\)
Let \(M'\) be the reflection of \(M\) in \(KL\). Since \(\Delta OLK \sim \Delta OST\), it suffices to prove that \(OM, OM'\) are isogonal lines wrt \(\angle O.\) Note that \(\angle MLO = \angle CAX = \angle BAY = \angle MKO\) so by isogonality lemma we are done.
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alex_g
26 posts
#9 • 3 Y
Y by a_friendwr_a, Adventure10, Mango247
Well, since no one bashed it by using Menelaus and Steiner yet, here it goes :

Notice that $KL$ and $ST$ both cut $BC$ in the exterior of segment $[XY]$, and denote by $U$ and $V$ these intersections. We have to prove that $U = V$, and because both $U$ and $V$ lie in the exterior of $[XY]$, it is enough to show that :

$\frac{UX}{UY} = \frac{VX}{VY},$ or after applying Menelaus' Theorem in $\triangle AXY$,

$\frac{KX}{KA} \cdot \frac{LA}{LY} = \frac{TX}{TA} \cdot \frac{SA}{SY},$ which rewrites as :

$\frac{BX}{CY} \cdot \frac{AC}{AB} \cdot \frac{cos \angle CAY}{cos \angle BAX} \cdot \frac{cos \angle AXB}{cos \angle CYA} = \frac{CX}{BY} \cdot \frac{AB}{AC} \cdot \frac{cos \angle BAY}{cos \angle CAX} \cdot \frac{cos \angle AXB}{cos \angle CYA}$

As the cosines cancel out by isogonality, this rewrites as :

$\frac{BX \cdot BY}{CX \cdot CY} = \frac{AB^2}{AC^2},$ which is true by Steiner's Theorem. Done.
This post has been edited 1 time. Last edited by alex_g, May 3, 2019, 8:23 PM
Reason: Error
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GorgonMathDota
1063 posts
#10 • 2 Y
Y by a_friendwr_a, Adventure10
fastlikearabbit wrote:
After inverting through $A$ with radius $\sqrt{AB\cdot AC}$ and then reflecting everything across the bisector of angle $\angle{BAC}$, you need this to finish: https://www.awesomemath.org/wp-pdf-files/math-reflections/mr-2015-05/article_1_lema_coaxalitate.pdf

After invert , then how ?
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silouan
3952 posts
#11 • 7 Y
Y by AlastorMoody, Hamel, Pathological, parmenides51, a_friendwr_a, khina, Adventure10
This problem was proposed by Greece and the author is Raphael Tsiamis.
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jbaca
225 posts
#12 • 3 Y
Y by microsoft_office_word, a_friendwr_a, Adventure10
Solution. A bit more detailed approach. Clearly, $ASKB$ and $ATLC$ are cyclic, from which we can obtain that $KSLT$ is also cyclic and notice that $TL\perp AB$ and $KS\perp AC$. Let $E$ be the midpoint of $\overline{BC}$, thus $EM\parallel AB$ and then $EM\perp TL$; hence, being $E$ the center of $(ATLC)$, we must have that $MT=ML$. Similarly, we get $MS=MK$. Since $K,\ S,\ L$ and $X$ all lie on the same circle, this forces $MS=MK=MT=ML$ to happen.

Let $H$ be the foot of the $A$-altitude. $H$ is the second intersection point of $(ASKHB)$ and $(AHTLC)$; therefore, applying the radical axis theorem to these circles and $(KSLT)$ we infer that $AH,\ SK$ and $TL$ concur at a point, say $P$.

Let $Z=\overline{ST}\cap \overline{KL}$. By Brockard's theorem, $MZ\perp AP$, so $H,\ Z$ and $M$ are collinear. We are done!
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Pathological
578 posts
#13 • 9 Y
Y by albind, Bassiskicking, parola, a_friendwr_a, Infinityfun, Wizard0001, Adventure10, Mango247, TheHimMan
Quick Pascal
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albind
28 posts
#14 • 3 Y
Y by a_friendwr_a, Adventure10, Mango247
Let $BK$$\cap$$AS$$=$$F$, $BS$$\cap$$AK$$=$$G$,$CL$$\cap$$AX$$=$$W$ and $CT$$\cap$$AY$$=$$M$.By simple angle chasing we find that $KSLT$ is cyclic. And the way we defined points gives us that $FKWL$,$MTGS$ are cyclics.And also $FKGS$ and $TWLM$ are cyclics.We use Power of point wrt to $FKGS$, $FKWL$ and $MTGS$ and we can deduce that B is a point in the radical axis of $FKWL$ and $MTGS$.WE use POP again wrt to $FKWL$, $MTGS$ and $TWLM$ and get that point $C$ is a point on the radical axis of the circles $FKWL$ and $MTGS$. So $BC$ is the radical axis of circles $FKWL$ and $MTGS$. Now use POP wrt to $KSLT$,$MTGS$ and $FKWL$ and get that $KL$, $TS$ and $BC$ are concurrent.
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iDra36
29 posts
#15 • 7 Y
Y by rmtf1111, Math_tricks, augustin_p, microsoft_office_word, a_friendwr_a, Adventure10, Mango247
Let $KS\cap TL =F$ , $BS\cap TC=D$ , $BK\cap CL=E$ . We will prove that points $D$ , $F$ and $E$ are collinear . Let $D'$ be the reflexion of $D$ over the midpoint of $TS$ . $ ATDS$ is cyclic because $\angle CTA=90$ and $\angle BSA=90$ . $D'$ is the orthocenter of $\Delta ATS$ because it's reflexion over the midpoint of $TS$ is $D$ . $AD$ and $AD'$ are isogonal in $\Delta ATS$ , because $AD$ is diameter of $(ATDS)$ . $\Delta DST$$ \sim $$\Delta EKL$ , because $\angle EKL=90-(90-\angle TSD)=\angle TSD$ and $\angle ELK=90-(90-\angle DTS)=\angle DTS$ . $\Delta TD'S\sim \Delta DST\sim \Delta EKL$ . $\Delta TFS \sim \Delta KFL$ $=>$$FTD'S \sim  EKFL$$=>$ $\angle EFL = \angle D'FS$ . If points $D$ , $F$ and $E$ are collinear then $\angle EFL =\angle DFT$ or $\angle D'FS =  \angle DFT$ . Now using trig Ceva two times in quadrilateral $ATFS$ , with point $D$ and with point $D'$ yields $\angle D'FS=\angle DFT=>$ points $D$ , $F$ and $E$ are collinear$=>$$BC$ , $TS$ and $KL$ are concurrent because of Desargues theorem in $\Delta BSK$ and $\Delta CTL$ .
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FISHMJ25
293 posts
#20 • 5 Y
Y by Pluto1708, Igore, a_friendwr_a, Adventure10, Mango247
After we invert around $A$ with power $\sqrt{bc}$ problem is equivalent to this:
Given triangle $ABC$ and let antipod of $A$ be $A'$.$X,Y$ are on $(ABC)$ (on shorter arc $BC$) s.t. $<BAX = <CAY $. $AX$ intersects $BA'$ and $CA'$ at $P$,$Q$, and $AY$ intersects $BA'$ and $CA'$ at $R$,$S$. Prove that center of spiral similarity that sends $QR$ to $PS$ lies on $(ABC)$.
First we compute $p$ as intersection of $b(-a)$ and $ax$. We get $$p=\frac{ax(b-a)+ba(a+x)}{ax+ba}$$or $$p=\frac{2bx+a(b-x)}{b+x}$$analogously $$q=\frac{2cx+a(c-x)}{c+x}$$(noticing that we just need to exchange $b$ with $c$ in intersection formula to get $q$ from $p$) And now we see that we will get $r,s$ by substituting $x$ with $\frac{bc}{x}$.So $$r=\frac{2bc+a(x-c)}{c+x}$$and $$s=\frac{2bc+a(x-b)}{b+x}$$And now the only real computation is finding $M$ , so we get $$m=\frac{qs-pr}{q+s-p-r}$$.Now we just bash, but first we do denominator because it seems easier. $$q+s=\frac{(2cx+a(c-x))(b+x)+(2bc+a(x-b))(c+x)}{(b+x)(c+x)}$$$$q+s=\frac{4xbc+2xa(c-b)+2cx^2+2bc^2}{(b+x)(c+x)}$$By symmetry (again we can just substitute $b$ with $c$ and vice versa) $$p+r=\frac{4xbc+2xa(b-c)+2bx^2+2cb^2}{(b+x)(c+x)}$$Finishing we get $$q+s-p-r=\frac{2(c-b)(x^2+bc+2xa)}{(b+x)(c+x)}$$Now we see something quite nice, we see the term $x^2+bc+2xa$ which will after conjugation become $\frac{2bcx+ax^2+abc}{x^2abc}$, so this part ($2bcx+ax^2+abc$) needs to be in the numerator if problem statement is true,or else after conjugation there will be no term equal to its conjugate(or it can be both in numerator and denominator so it cancels).It is hard for me to explain this part but if we have $\frac{t}{q}$ and we wish to show that it lies on unit circle we need to have $\frac{q}{t}=\frac{\bar t}{\bar q}$ then $q=\bar t k$ with $k$ having modulus $1$. This way if we have trouble factoring we can notice that on of factors will be $2bcx+ax^2+abc$(also it is safe to assume one of them will be $b-c$). Here is the harder part: $$qs=\frac{(2cx+a(c-x))(2bc+a(x-b))}{(b+x)(c+x)}$$or $$qs=\frac{4c^2bx+2acx(x-b)+2abc(c-x)+a^2(c-x)(x-b)}{(b+x)(c+x)}$$and again by symmetry $$pr=\frac{4b^2cx+2abx(x-c)+2abc(b-x)+a^2(b-x)(x-c)}{(b+x)(c+x)}$$$$qs-pr=\frac{2(c-b)(2bcx+ax^2+abc)}{(b+x)(c+x)}$$and finaly $$m=\frac{qs-pr}{q+s-p-r}=\frac{2bcx+ax^2+abc}{x^2+bc+2xa}$$and we see $m=\frac{1}{\bar m}$ which finishes problem.
Motivation: We want to bash, thus we invert after seeing that is is immpossible in given problem and that there are isogonal conjugates. Now, $p$ is not that bad,and by symmetry we can get others. And finally we are able to handle $m$ because $M$ is center of spiral similarity. We see that it is possible to bash this problem,which leads to solution above. (note:this should be the method of last resort in a competition,but its nice for practicing bashing). This takes about 45min-1h.
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Pure_IQ
426 posts
#21 • 5 Y
Y by AlastorMoody, parola, a_friendwr_a, Adventure10, Mango247
Sorry for reviving, but tought i'd share this Similarity based solution, as nobody seems to have mentioned it.

The result is equivalent with: $(AXKT)=(AYLS)$, which follows from:
- $\triangle AKB \sim \triangle ALC$ ;
- $\triangle ASB \sim \triangle ATC$ ;
- $\frac{BK}{TC}=\frac{BX}{XC}$ and $\frac{BS}{CL}=\frac{BY}{YC}$ ; (Thales)
This post has been edited 1 time. Last edited by Pure_IQ, Sep 8, 2019, 8:56 PM
Reason: Forgot to mention the cross-ratios :)
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math_pi_rate
1218 posts
#22 • 6 Y
Y by amar_04, sameer_chahar12, itslumi, a_friendwr_a, Iora, Adventure10
Here's a solution using moving points: Fix $\triangle ABC$, and animate $X$ linearly on $BC$. By isogonality, $X \mapsto Y$ is a projection. Also, note that $K,S$ simply lie on the circle with $AB$ as diameter and $L,T$ lie on the circle with $AC$ as diameter. Thus, by perspectivity from $A$, we get that $X \mapsto K,T$ is a projective map. Similarly, $Y \mapsto L,S$ is also a projective map. This gives that the degree of each of the points $K,L,S,T$ is two. Now, when $X=A \infty_{\perp_{AB}} \cap BC$, it's easy to see that $Y=A \infty_{\perp_{AC}} \cap BC$ (since, in this case, $\angle XAB=\angle YAC=90^{\circ}$). Then, in this situation, we will have $K=L=A$ (as $AX$ is tangent to $\odot (AB)$ and $AY$ is tangent to $\odot (AC)$). This gives that the degree of line $KL$ is atmost $2+2-1=3$. Similarly, we have that degree of $ST$ is also atmost $3$. Thus it suffices to show that $KL,ST,BC$ are concurrent for $(3+3+0)+1=7$ positions of $X$.
  • When $X$ is the foot of the $A$-altitude, we have $K=S=X$, which directly gives the desired result.
  • When $X$ is taken so that $Y$ is the foot of the $A$-altitude, we again have $L=T=Y$, and hence the result.
  • For $X=B$, we get $Y=C$. This gives $K=B$ and $L=C$, and so $KL,BC$ only coincide.
  • The case $X=C$ and $Y=B$ follows similarly as above.
  • Take $X$ as the foot of the $A$-internal angle bisector. Then $Y=X$, and we have that lines $KL,ST$ themselves coincide.
  • The case when $X$ is the foot of the $A$-external angle bisector follows in a similar fashion as above.
  • The seventh position is a bit of a problem, as I cannot seem to find a nice position which would not require any solving per se. For now, you can consider any point you like :D ( I'd recommend $X=\infty_{BC}$). I'll try to find a better point later on.
This post has been edited 2 times. Last edited by math_pi_rate, Jan 10, 2020, 10:17 AM
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Implicit25
1 post
#23 • 2 Y
Y by a_friendwr_a, Adventure10
Dear mathlinkers, I don't know if I'm doing it right or not. Can you check it for me, please!
Here is my solution:
Let H is the feet of A on BC.
We have: (HT,HA)= (CT,CA) (since (A,T,H,C) cyclic))= π/2 - (AC,AT)=π/2-(AC,AX)
=π/2-(AY,AB) (since aX,AY are isogonal in angle BAC)
=π/2-(AS,AB)= (BA,BS)= (HA,HS) (because (A,S,H,B) cyclic) (mod π).
Then HT, HS are reflect about HA.
On the other hand: (HK,HB)=(AK,AB)=(AX,AB)=(AC,AY)=(AC,AL)=(HC,HL).
So that HK, HL are reflect about HA.
Now we use double ratio of division on circle:
$\odot (AB)$ (AB,KS)= H(AB,KS)= H(AC,LT) (since pencil reflection) = $\odot (AC)$ (AC,LT) (?! :read: )
So that (AB,KS)=(AC,LT) as known as BC, KL, ST converge.
This post has been edited 1 time. Last edited by Implicit25, Jan 30, 2020, 7:27 PM
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kapilpavase
595 posts
#24 • 5 Y
Y by a_friendwr_a, Illuzion, Adventure10, Mango247, axolotlx7
Pretty straightforward by cross ratio.
We have $\dfrac{LY}{SY}=\dfrac{CL}{BS}$ and $\dfrac{LA}{SA}=\dfrac{CL \cot \angle{CAY}}{BS \cot \angle{BAY}}$. So that $\{L,S;Y,A\}=-\dfrac{ \cot \angle{BAY}}{\cot \angle{CAY}}$. Similarly $\{K,T;X,A\}=-\dfrac{\cot \angle{CAX}}{\cot \angle{BAX}}$. Since $\angle{BAX}=\angle{CAY}$ we get that cross ratios are equal and hence $ST,XY,KL$ concur. Done
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GeoMetrix
924 posts
#25 • 2 Y
Y by mueller.25, a_friendwr_a
Child's play for desergaues.

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[/asy]

Proof: Note that by desergaues theorem we just need to prove that $\Delta NBX$ and $\Delta YCM$ are perspective with respect to a line or that if $H=BN \cap CY,I=BX\cap CM$ then $A-H-I$ are collinear.Let $K=BY \cap CN$ Firstly observe that since $\angle ANK+\angle AYK\equiv \angle ANC +\angle AYB=90+90=180$ we have that $ANKY$ is cyclic. Similiarly since $\angle AXI+\angle AMI=360-(\angle AXB+\angle AMC)=180$ we have $AXMI$ is cyclic .Note that by isogonal line lemma we have that $AH,AK$ are isogonal. So we just need to prove $AI,AK$ as isogonal. Now note that $\angle BAX=\angle MAC$ since $AP,AQ$ are isogonal and also $\angle BXA=\angle CMA=90$ so by $AA$ similiarity we have$$\Delta ABX\sim \Delta ACM\implies \frac{AB}{AC}=\frac{AX}{AM}.......(1)$$. Also note that as $\angle NAC\equiv \angle PAC=\angle QAB \equiv \angle YAB$ since $AP,AQ$ are isogonal and also as $\angle AYB=\angle ANB=90$ so by $AA$ similiarity we have$$\Delta AYB \sim \Delta ANC \implies \frac{AY}{AN}=\frac{AB}{AC}....(2)$$Now (1), (2) imply that $\Delta AYN \sim \Delta AXM$ by $SAS$ similiarity $\implies \angle ANY=\angle AMX$ and since $AKNY$ and $AXIM$ are cyclic we have $\angle AKY=\angle ANY=\angle AMX=\angle AIX \implies \angle YAK=90-\angle AKY=90-\angle AIX=\angle IAX \implies AI,AK$ are isogonal w.r.t $\angle PAQ \implies AI,AK$. Now note that by the isogonality we obtained we have $\angle BAK=\angle BAP+\angle XAK=\angle CAM+\angle MAO = \angle CAM\implies AK,AI $ are isogonal w.r.t $\angle BAC$ . We are done. $\blacksquare$
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Eyed
1065 posts
#27 • 2 Y
Y by albgeo, a_friendwr_a
Menelaus solution:
Let $P = LK\cap BC$. By menelaus, we have $\frac{AL}{LY} \cdot\frac{YP}{PX}\cdot\frac{XK}{KA} = -1$. If $S,T,P$ were collinear, then we would have $\frac{AS}{SY}\cdot\frac{YP}{PX}\cdot\frac{XT}{TA} = -1$, thus we need to prove that these two expressions are equal. Setting these equal to each other and simplifying, we have to prove that $$\frac{AL}{LY}\cdot\frac{XK}{KA} = \frac{XT}{TA}\cdot\frac{AS}{SY}$$Let $\angle BAY = \angle CAX = \gamma$ and $\angle YAX = \alpha$. Then, since $\angle ALC = 90$, we have $AL = AC\cos(\alpha+\gamma)$. Similarly, we also have $KA = AB\cos(\alpha+\gamma)$, thus the LHS of what we want to prove becomes $\frac{AC}{LY}\cdot\frac{XK}{AB}$. In a similar vein, the RHS becomes $\frac{AB}{SY}\cdot \frac{XT}{AC}$.
Since $BK$ is parallel to $CT$, we let $\angle KBC = \angle BCT = \theta$ and $\angle CBS = \angle BCL = \mu$. Then, we have $$LY = CY\sin(\mu), XK = BX\sin(\theta), XT = XC\sin(\theta), SY = BY\sin(\mu)$$Thus, plugging this is, our equation becomes
$$\frac{AC}{AB} \cdot \frac{BX\sin(\theta)}{CY\sin(\mu)} = \frac{AB}{AC}\cdot\frac{CX\sin(\theta)}{BY\sin(\mu)}$$$$\Leftrightarrow \frac{AC}{AB}\cdot\frac{BX}{CX} = \frac{AB}{AC}\cdot\frac{CY}{BY}$$However, by the ratio lemma, $\frac{BX}{CX} = \frac{AB}{AC}\cdot\frac{\sin(\gamma+\alpha)}{\sin(\gamma)}$ and $\frac{CY}{BY} = \frac{AC}{AB}\cdot\frac{\sin(\gamma+\alpha)}{\sin(\gamma)}$. Plugging this in our equation becomes
$$\frac{AC}{AB}\cdot\frac{AB}{AC}\cdot\frac{\sin(\gamma+\alpha)}{\sin(\gamma)}= \frac{AB}{AC}\cdot\frac{AC}{AB}\cdot\frac{\sin(\gamma+\alpha)}{\sin(\gamma)}$$$$\Leftrightarrow 1 = 1$$Since all of our steps are reversible, we have proven that the original expression is true. Thus, $S,T,P$ are collinear, so $KL$ and $ST$ must intersect at some point on $BC$.
This post has been edited 1 time. Last edited by Eyed, Apr 1, 2020, 12:35 AM
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thegameisover
10 posts
#28 • 1 Y
Y by a_friendwr_a
Flash_Sloth wrote:
Furthermore, let $M$ be the midpoint of BC, then by simple reflexion argument $ML =MS$ and $MT =MK$, which means $M$ is the center of $\odot (LSKT)$.
Where do $ML=MS$ and $MT=MK$ come from?
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Kamran011
678 posts
#30 • 1 Y
Y by a_friendwr_a
First by some angle chasing (or trig) it's easy to see $KLST$ is cyclic. Let $KS\cap LT=P$ and, consider the Miquel point $M$ of $KLST$ ,and by Brokard theorem $\angle PMA=90$ .
Now invert around $A$ with radius $\sqrt{AS.AL}$ , let $X'$ denote the inverse of some point $X$ .Then
$\angle AC'P=\angle AC'S=\angle ALC=90$ , and similarly $\angle AB'P=90$ and so $\{ C',B'\}$ lie on the circle with diameter $AP$ , inverting back we're done. $\blacksquare$
This post has been edited 4 times. Last edited by Kamran011, Aug 20, 2020, 7:18 AM
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Ali3085
214 posts
#35 • 3 Y
Y by Math-48, Muaaz.SY, a_friendwr_a
Is this really $G6$ it must be $G1$ or $G2$
let $M,N,P$ the midpoints of $BC,AB,AC$
let $KL \cap ST=Z$ and $KS \cap LT=W$
main calim(2): $KLST$ si cyclic
$\angle ASK=\angle ABK=90-\angle BAK=90-\angle CAS$ so $SK \perp AC$
so $MN \perp SK$ since $NA=NB=NK=NS$ so $MS=MK$
in a similar way $ML=MT$ it's obvious that $ML=MS$ so $ML=MS=MK=MT$
$\blacksquare$
radical axis on $(AKSB),(ALTC),(KLST)$ we have $W$ is on the altitude from $A$ to $BC$
Brokard on $KLST$ gives $AW \perp MZ$ so $Z \in BC$
This post has been edited 1 time. Last edited by Ali3085, Jan 8, 2021, 8:04 AM
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Muaaz.SY
90 posts
#36 • 2 Y
Y by a_friendwr_a, Math-48
Bro it's even easy for G2
Note that ignoring trivial special cases we could $WLOG$ suppose that $X$ is between $Y$ and $C$ and $T,S$ are outside $\triangle ABC$.
Let $KL\cap BC=M_1 , ST\cap BC=M_2$
It's easy to se by Menelaus and law of sins that
$\frac{XM_1}{M_1Y}=\frac{YL}{LA}.\frac{AK}{KX}$

$=\frac{CY.cos(\angle{BAY}+\hat{B})}{AC.cos\angle{CAY}}.\frac{AB.cos\angle{BAX}}{BX.cos(\angle{CAX}+\hat{C})}$

$=\frac{AB.CY}{AC.BX}.\frac{cos(\angle{BAY}+\hat{B})}{cos(\angle{CAX}+\hat{C})}$

Also $\frac{XM_2}{M_2Y}=\frac{YS}{SA}.\frac{AT}{TX}
=\frac{BY.AC}{CX.AB}.\frac{cos(\angle{BAY}+\hat{B})}{cos(\angle{CAX}+\hat{C})}$

So $\frac{XM_1}{M_1Y}=\frac{XM_2}{M_2Y} \Leftrightarrow \left( \frac{AB}{AC}\right)^2=\frac{BX}{XC}.\frac{BY}{YC}$

Which is a well known relation so $M_1=M_2$ , done
This post has been edited 2 times. Last edited by Muaaz.SY, Jan 10, 2021, 7:53 PM
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BarisKoyuncu
577 posts
#37 • 2 Y
Y by Mango247, Mango247
Simple angle chasing gives us that $KTLS$ is cyclic. Let $H$ be the foot of the $A$-altitude. Since $AKSH$ and $ATLH$ are cyclic, we can see that $H$ is the Miquel Point of the cyclic quadrilateral $KTLS$. Let $KL\cup ST=E$. From a well-known lemma, $\angle APE=90^\circ$. Hence, $E$ lies on $BC$.
This post has been edited 1 time. Last edited by BarisKoyuncu, Aug 30, 2021, 9:50 AM
Reason: .
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Ru83n05
170 posts
#38
Y by
Let $H$ be the foot of the $A$ altidue. By angle chasing:

Lemma: $BC$ is the external bisector of both $\angle SHL$ and $\angle KHT$

So both $SL\cap BC$ and $KT\cap BC$ are the point $P$ such that $(X, Y; H, P)=-1$, the end. $\blacksquare$
Remark: Alternatively, after showing that $KTLS$ is cyclic, you can prove the result by fixing $\triangle ALS$ and animating $T\in AX$ linearly (the condition has degree 4, which is now easy).
This post has been edited 2 times. Last edited by Ru83n05, Feb 6, 2022, 12:34 PM
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Mahdi_Mashayekhi
695 posts
#40
Y by
Let $AH$ be altitude and $KL, ST$ meet at $P$.
Claim1 : $LKTS$ is cyclic.
Proof : Let $\angle BAY = \angle CAX = x$ and $\angle BAX = \angle CAY = y$. $AL.AS = b.cos{y} . c.cos{x} = AK.AT$ so $LKTS$ is cyclic.

Note that $AKHS$ and $ALHT$ are cyclic so $AKS$ and $ALT$ meet at $H$ and it's well known that $\angle AHP = \angle 90$ so $P$ lies on $BC$.
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Akkuman
12 posts
#41
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BY SIMPLE ANGLE CHASING, WE GET LKST IS CYCLIC
THEN BY RADICAL AXIS THEOREM, AD( LET THIS BE THE A-ALTITUDE OF ABC WITH D ON BC), LT AND KS ALL ARE CONCURRENT. ( LET THIS MEETING POINT BE P)
LET F=LK U ST AND LET AP U ST =E
THEN WE KNOW (F, E, S, T) MUST BE HARMONIC
ALDTC AND AKDSB ARE BOTH CYCLIC, THUS ANGLE ADK=ANGLE ABK = ANGLE ACL = ANGLE ADL
THUS AD BISECTS ANGLE LDK .
LET LK U AD = O AND LET BC U LK = Q
WE GET (Q, O, L, K) IS HARMONIC ( TRIVIAL)
SINCE ALS, ADE AND AKT ARE ALL COLLINEAR,
THUS A ,Q, AND F MUST BE COLLINEAR, BUT SINCE A DOES NOT LIE ON LK,
THERE IS ONLY ONE POSSIBILITY THAT Q=F.

HENCE PROVED :)
This post has been edited 1 time. Last edited by Akkuman, May 16, 2022, 8:28 AM
Reason: WRONG NAME FOR POINT CHOSEN
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ike.chen
1162 posts
#42
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Let $P = KL \cap ST$. Some basic angle chasing yields $ABSK \sim ACTL$. Thus, $$\angle LSK = 180^{\circ} - \angle ASK = 180^{\circ} - \angle ATL = 180^{\circ} - \angle LTK$$which means $KSLT$ is cyclic. Now, parallel lines, Steiner's Ratio Theorem, and similarity give $$\frac{LY}{YS} \cdot \frac{TX}{XK} = \frac{CY}{YB} \cdot \frac{CX}{XB} = \left( \frac{AC}{AB} \right)^2 =  \left( \frac{TL}{SK} \right)^2 =  \left( \frac{PL}{PS} \right)^2.$$Thus, properties of anti-parallel lines and Phantom Points imply $P, X, Y$ are collinear, which clearly finishes. $\blacksquare$


Remark: Some details are left out of this write-up, as I am short on time.

Also, how did I miss the Brocard's solution after noticing that Radical Axes imply $KS$ and $LT$ meet on the $A$-altitude?
This post has been edited 1 time. Last edited by ike.chen, Aug 29, 2023, 5:46 AM
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Aiden-1089
285 posts
#43
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Let $P=KL \cap ST$.
Firstly, note that $$AS \cdot AL = (AB \cos{\angle BAY})(AC \cos{\angle CAX}) = (AB \cos{\angle BAX})(AC \cos{\angle CAY})=AK \cdot AT,$$so points $K,S,T,L$ are concyclic by pop. Let $O$ be the circumcenter of $(KSTL)$.
Next, consider the radical centre $Q$ of circles $(ABKS), (ACTL), (KSTL)$. Let $A'$ be the foot of the $A$-altitude of $\Delta ABC$.
Since points $A$ and $A'$ lie on circles $(ABKS)$ and $(ACTL)$, $Q=KS \cap TL$ lies on the $A$-altitude of $\Delta ABC$. By Brocard's theorem, it suffices to show that $O$ lies on $BC$.
Let $\perp_{UV}$ denote the perpendicular bisector of segment $UV$, so that $O=\perp_{KS} \cap \perp_{TL}$.
Let $D=SK \cap BC, E=SK \cap AC$, and let $M$ be the midpoint of $BC$.
$\measuredangle KDB = \measuredangle KBD + \measuredangle DKB = \measuredangle KAA' + \measuredangle YAB = \measuredangle XAA' + \measuredangle CAX = \measuredangle CAA'$, so $\measuredangle DEC = \measuredangle EDC + \measuredangle DCE = \measuredangle CAA' + \measuredangle A'CA = \measuredangle CA'A = 90^{\circ}$.
It follows that $KS \perp AC$. Consider circle $(ABKS)$, since $\perp_{KS}$ passes through the midpoint of $AB$ and is parallel to $AC$, it passes through $M$.
Similarly, $\perp_{TL}$ passes through $M$. It follows that $O=M$, so we are done. $\square$
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axolotlx7
133 posts
#44 • 1 Y
Y by Aiden-1089
The statement after taking pole-polar duality at $A$ with arbitrary radius is quite interesting (and arguably easier). Let $\ell_P$ denote the polar of $P$, and $P_{\ell}$ denote the pole of $\ell$.

Obtaining the dual statement. Note that $\ell_X, \ell_Y, \ell_B, \ell_C$ all pass through $P_{BCXY}$ with $(\ell_X, \ell_Y)$ being equally inclined with respect to $(\ell_B, \ell_C)$ (basically $\measuredangle (\ell_B, \ell_X)=\measuredangle (\ell_Y, \ell_C)$). Now
\[ K \in AX, AX \perp KB \Leftrightarrow P_{AX} \in \ell_K, AP_{AX} \perp AP_{KB} \]which can be rephrased to $\ell_K \parallel \ell_X$ and $\ell_X \perp AP_{KB}$. Basically, $P_{KB}$ is the point on $\ell_B$ such that $AP_{BK} \perp \ell_X$. The other points are obtained in a similar manner. The finished diagram is as shown, and we want to show that $P_{KL}, P_{BCXY}, P_{ST}$ are collinear.
https://i.imgur.com/sV3o9RM.png
Solving the problem. Note that
\[ \measuredangle(\ell_B) + \measuredangle(\ell_C) = \measuredangle(\ell_X) + \measuredangle(\ell_Y) = \measuredangle(\overline{AP_{BK}P_{CT}}) + \measuredangle(\overline{AP_{CL}P_{BS}}) \]implies $P_{BK}P_{CL}P_{BS}P_{CT}$ is cyclic (and allows us to delete $\ell_X$ and $\ell_Y$ from the diagram). Let $O$ be the center of the circle. Let $P_{BK}P_{KL}$ and $P_{CL}P_{KL}$ meet the circle again at $M$ and $N$ respectively. Then $P_{CT}M$ and $P_{BS}N$ are diameters, so they intersect at $O$. Hence by Pascal on $P_{CT}P_{CL}NP_{BS}P_{BK}M$, we get $O, P_{BCXY}, P_{KL}$ collinear. Similarly, $O, P_{BCXY}, P_{ST}$ collinear, so the result follows.
This post has been edited 3 times. Last edited by axolotlx7, Apr 2, 2024, 10:01 AM
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TestX01
341 posts
#45 • 1 Y
Y by GeoKing
why 3 n on imo. anyways geo grind never stops: original solution.

Let $P=XL\cap YK$. It suffices to prove that $P$ lies on the $A$-altitude, as then, letting $D$ be the foot of $A$ to $BC$, by complete quad we have $(KL\cap BC,D;X,Y)=-1$, and a similar result for $ST$, and since $D$ is fixed we can get the concurrence being shared.

We use Ceva. Let $\angle XAB=\alpha,\angle BXA=\beta, \angle AYC=\theta$. Clearly,
\[\frac{XK}{KA}\times\frac{AL}{LY}\times \frac{YD}{DX}=\frac{\tan \alpha}{\tan \beta}\times \frac{\tan \theta}{\tan \alpha}\times \frac{\tan \beta}{\tan \theta}=1\]from the right triangles, and isogonality.
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khanhnx
1618 posts
#46 • 1 Y
Y by GeoKing
Suppose that $M, N, P$ are midpoint of $BC, CA, AB;$ $KS$ intersects $TL$ at $U;$ $KL$ intersects $TS$ at $V$. It's easy to see that $K, S \in \bigodot(AB)$ and $L, T \in \bigodot(CA)$. We have $\angle{ASK} = 180^{\circ} - \angle{ABK} = 180^{\circ} - \angle{ACL} = \angle{ATL}$. Then $K, S, L, T$ lie on a circle. We also have $\angle{AKS} = \angle{ABS} = \angle{ACT} = 90^{\circ} - \angle{CAX}$. Hence $KS \perp CA$ or $KS \perp MN$. Similarly, $LT \perp MP$. Then $M$ is center of $(KSLT)$. Note that the radical axis of $\bigodot(AB)$ and $\bigodot(CA)$ is $A$ - altitude of $\triangle ABC$ and $\mathcal{P}_{U / \bigodot(AB)} = \overline{UK} \cdot \overline{US} = \overline{UL} \cdot \overline{UT} = \mathcal{P}_{U / \bigodot(CA)},$ we have $AU \perp BC$. But by Brocard theorem, $MV \perp AU$ then $V \in BC$
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eulerleonhardfan
53 posts
#47
Y by
Quite a nice and simple problem, with a bit of bashing lol.

Notice that $$\angle AXY = \angle ABY = 90 - \angle BAY = 90 - \angle CAN = \angle ACN = \angle AMN$$so $XYMN$ is cyclic, we get $\frac{\cos \angle NYM}{\cos \angle XNY} = \frac{\cos \angle NXM}{\cos \angle XMY}$ and notice that $\triangle BXY \sim \triangle CMN$, so $\frac{BY}{CN} = \frac{BX}{CM}$. Combining the two gives $$\frac{BY}{CN} \cdot \frac{\cos \angle NYM}{\cos \angle XNY} = \frac{BX}{CM} \cdot \frac{\cos \angle NXM}{\cos \angle XMY}$$$$\Rightarrow \frac{BY}{CN} \cdot \frac{\sin \angle BYN}{\sin \angle CNY} = \frac{BX}{CM} \cdot \frac{\sin \angle BXM}{\sin \angle CMX} \quad (\dag)$$Let $D = \overline{BC} \cap \overline{NY}, E = \overline{BC} \cap \overline{MX}$, we have $$\frac{BD}{\sin \angle BYD} \cdot \frac{\sin \angle CND}{CD} = \frac{BY}{\sin \angle BDY} \cdot \frac{\sin \angle CDN}{CN} = \frac{BY}{CN}$$$$\Rightarrow \frac{BD}{CD} = \frac{BY}{CN} \cdot \frac{\sin  \angle BYN}{\sin \angle CNY}$$Similarly, $$\frac{BE}{CE} = \frac{BX}{CM} \cdot \frac{\sin \angle BXM}{\sin \angle CMX}$$And by $(\dag)$, we have $\frac{BD}{CD} = \frac{BE}{CE}$, i.e. $D = E$, so $\overline{BC}, \overline{NY}, \overline{MX}$ are concurrent, as desired.
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anirbanbz
24 posts
#48
Y by
$\textbf{This was way too easy for a G6/P3}$

$\text{Define}$ $\omega_{B}$ $\text{and}$ $\omega_{C}$ $\text{as the circles with diameter}$ $AC$, $\text{and}$ $AB$, $\text{respectively}$.
$\newline$
$\text{Further define}$ $D$ $\text{as the foot of the perpendicular from }$ $A$
$\newline$
$\text{Lastly, define}$ $X=KS\cap LT$, $\text{and}$ $Y=KL\cap TS$
$\newline$
$\text{Now, obviously}$ $K, S, D \in \omega_{C},$ $\text{and}$ $L, T, D \in \omega_{B}$
$\newline$
$\textbf{Claim:}$ $K,L,S,T$ $\text{are cyclic}$
$\newline$
$\textbf{Proof:}$ $\angle BSK = \angle BAK = \angle CAL = \angle LTC$
$\angle LTK=\frac{\pi}{2}+\angle LTC=\frac{\pi}{2}+\angle BSK=\pi -(\frac{\pi}{2}-\angle BSK)=\pi-\angle LSK$
$\newline$
$\textbf{Claim:}$ $X$ $\text{lies on the radical axis of}$ $\omega_{B}$ $\text{and}$ $\omega_{C}$
$\newline$
$\textbf{Proof:}$
$\newline$
$\text{Radical axis theorem on }$ $\omega_{B},$ $\omega_{C}$ $\text{and}$ $(KSLT)$ $\text{suffices.}$
$\newline$
$\textbf{Finishing:}$
$\newline$
$D$ $\text{must be the Miquel point of the complete quadrilateral}$ $\mathcal{Q} \{KT,TL,LS,SK\},$ $\text{as}$ $D=(ADTLC)\cap (ADSKB)$
$\text{It is well-known that the Miquel point of a convex, complete, cyclic quadrilateral is the foot of the perpendicular}$
$\text{ from the intersection of the diagonals to the line connecting the intersections of the sides}$
$\newline$
$\Longrightarrow$ $YD\perp AX$.
$\newline$ $\text{But this finishes the problem, as}$ $\overline{AX}$ $\text{is the same as}$ $\overline{AD}$ $\text{and}$ $AD\perp BC$
$\newline$
$\Longrightarrow$ $Y\in BC,$ $\text{which is exactly what we wanted.}$ $\blacksquare$
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L13832
268 posts
#49 • 1 Y
Y by alexanderhamilton124
Main claim is to use Desargue's Theorem on $\triangle BSK$ and $\triangle CTL$ for $BC, TS$ and $KL$ concurrence. For this one needs to show $\overline{BK\cap CL-LT\cap SK-BS\cap CT}$ is collinear.
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Nari_Tom
117 posts
#50
Y by
Main claim is to use Desargue's theorem on triangles $BTK$ and $CSL$. So we just need to show that $D=BT\cap CS$, $E=BK\cap CL$ and $A=KT \cap SL$ are collinear. If we denote $F=BS \cap CT$ by the parallelogram isogonolaty lemma $AD, AF$ and $AE, AF$ are isogonal as we wanted.
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Nari_Tom
117 posts
#51
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Nari_Tom wrote:
Main claim is to use Desargue's theorem on triangles $BTK$ and $CSL$. So we just need to show that $D=BT\cap CS$, $E=BK\cap CL$ and $A=KT \cap SL$ are collinear. If we denote $F=BS \cap CT$ by the ISOGONAL LINES lemma $AD, AF$ and $AE, AF$ are isogonal as we wanted.
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SimplisticFormulas
116 posts
#52 • 1 Y
Y by radian_51
i managed to find a way without desargues theorem, but the end part does use projective
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fearsum_fyz
52 posts
#53
Y by
Add in the circles with diameters $AB$ and $AC$. Let $D$ be the foot from $A$ and let $M$ be the midpoint of $BC$.

Claim: $K, L, S, T$ are concyclic with center $M$.
Proof. $\measuredangle{SLT} = \measuredangle{CLT} - 90^{\circ} = \measuredangle{CAT} - 90^{\circ} = \measuredangle{SAB} - 90^{\circ} = \measuredangle{SKB} - 90^{\circ} = \measuredangle{SKT}$
It is easy to see that $M$ must be the center since it lies on the perpendicular bisectors of $KT$ and $LS$.

Claim: $S, T, M, D$ and $K, L, M, D$ are also concyclic.
Proof. More angle chasing!
$\measuredangle{MDS} = \measuredangle{BDS} = \measuredangle{BAS} = \measuredangle{TAC} = \measuredangle{TDC} = \measuredangle{TDM}$ so $S, T, M, D$ are concyclic.
Similarly, $K, L, M, D$ are concyclic.

Finally, by radical axis concurrence on $(KLST)$, $(STMD)$, and $(KLMD)$, we have that $KL$ and $ST$ intersect on line $BC$.

https://i.imgur.com/v3eZP9h.png
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CatinoBarbaraCombinatoric
107 posts
#54
Y by
We want to prove that
$$(A,X;T,K)=(A,Y;S,L)$$Let $X'$ the intersection of the line perpendicural to $AX$ trought $A$ with $BC$ and define similarly $Y'$.
By projection respect infinity point of $AX'$ we have
$(A,X;T,K)=(X',X;C,B)=(AX',AX;AC,AB)$
Similarly
$(A,Y;S,L)=(AY',AY;AB,AC)$
The reflection respect angle bisector of $\angle BAC$ sends $AB$ in $AC$, $AX'$ in $AY'$ and $AX$ in $AY$.
So $(AY',AY;AB,AC)=(AX',AX;AC,AB)$ and $(A,X;T,K)=(A,Y;S,L)$ as we wanted.
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