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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
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0 replies
jlacosta
May 1, 2025
0 replies
Cool Functional Equation
Warideeb   1
N 4 minutes ago by maromex
Find all functions real to real such that
$f(xy+f(x))=xf(y)+f(x)$
for all reals $x,y$.
1 reply
+1 w
Warideeb
an hour ago
maromex
4 minutes ago
Functional equation
socrates   10
N 10 minutes ago by MathLuis
Source: Inspired by another
Determine all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that \[ \forall x, y \in \mathbb{R}^+ \ , \  \ f(x+f(xy))=f(x)+xf(y).\]
10 replies
socrates
Oct 28, 2014
MathLuis
10 minutes ago
Find f
Redriver   7
N 20 minutes ago by aaravdodhia
Find all $: R \to R : \ \ f(x^2+f(y))=y+f^2(x)$
7 replies
Redriver
Jun 25, 2006
aaravdodhia
20 minutes ago
IMO Shortlist 2011, G5
WakeUp   72
N 23 minutes ago by ItsBesi
Source: IMO Shortlist 2011, G5
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent.

Proposed by Irena Majcen and Kris Stopar, Slovenia
72 replies
WakeUp
Jul 13, 2012
ItsBesi
23 minutes ago
No more topics!
angles in triangle
AndrewTom   34
N May 13, 2025 by happypi31415
Source: BrMO 2012/13 Round 2
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.
34 replies
AndrewTom
Feb 1, 2013
happypi31415
May 13, 2025
angles in triangle
G H J
Source: BrMO 2012/13 Round 2
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AndrewTom
12750 posts
#1 • 4 Y
Y by Adventure10, Mango247, Rounak_iitr, ItsBesi
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.
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Nilashis
132 posts
#2 • 3 Y
Y by AndrewTom, Adventure10, Mango247
Use sin rule on $\triangle ABQ$, $\triangle ACQ$, $\triangle BAP$, $\triangle CAP$ and comparing them we get that
$\frac{sin\angle BAQ}{sin\angle QAC}=\frac{sin\angle CAP}{sin\angle PAB}$. Now take $\angle BAQ=x$ and $\angle PAC=y$ then the equation reduces to $\frac{sinx}{sin(A-x)}=\frac{siny}{sin(A-y)}$
$2sin(A-x)siny=2sin(A-y)sinx$
$cos(A-x-y)-cos(A-x+y)=cos(A-y-x)-cos(A-y+x)$
$cos(A-x+y)=cos(A-y+x)$
$x-y=y-x$
$x=y$
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nsato
15655 posts
#3 • 2 Y
Y by Adventure10, ehuseyinyigit
This appears as an exercise in Geometry Revisited (Section 1.9, Exercise 3).
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sunken rock
4401 posts
#4 • 3 Y
Y by AndrewTom, Adventure10, Mango247
It has been posted here around few years ago, with a very nice synthetic solution!

Best regards,
sunken rock
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MMEEvN
252 posts
#5 • 12 Y
Y by sunken rock, AndrewTom, kprepaf, jlammy, 93051, v_Enhance, Med_Sqrt, hakN, Adventure10, Mango247, ohiorizzler1434, ehuseyinyigit
Let $R$ be the point such that $APBR$ is a parallelogram . Hence $AR || BP ||QC$ and $AR=BP=CQ$ Hence $ARQC$ is a parallelogram.$\angle ACQ = \angle ARQ$ . But $ \angle ACQ = \angle ABQ$ . Hence $ARBQ$ is cyclic.
.$\angle PAB=\angle ABR =\angle AQR= \angle QAC$. $ \Longrightarrow \angle QAB=\angle PAC$
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jlammy
1099 posts
#6 • 2 Y
Y by Adventure10, Mango247
sunken rock wrote:
It has been posted here around few years ago, with a very nice synthetic solution!

Best regards,
sunken rock

Can you specify the details of this "very nice synthetic solution"?
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sunken rock
4401 posts
#7 • 2 Y
Y by Adventure10, Mango247
@jlammy: Like MMEEvN did!

Best regards,
sunken rock
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IDMasterz
1412 posts
#8 • 1 Y
Y by Adventure10
no angle chasing: Let $P'$ be the $P$ isogonal conjugate and $P''$ be its reflection over $BC$. The angle bisectors of $BPC$ and $BAC$ are obviously parallel. $AP, PP''$ are antiparallel wrt $BPC$ so $AP' \parallel PP'' \parallel QP'$ since $PP'QP''$ form a parallelogram, so done.
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DottedCaculator
7357 posts
#9
Y by
Solution
This post has been edited 1 time. Last edited by DottedCaculator, Dec 10, 2021, 10:45 PM
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827681
163 posts
#11
Y by
AndrewTom wrote:
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.

For a non complicated solution unlike above here's a hint
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guptaamitu1
657 posts
#12 • 1 Y
Y by Rounak_iitr
Here's a different proof with similar triangles and homothety (plus reflection in angle bisector)
Let $E = \overline{BP} \cap \overline{AC}, F = \overline{CP} \cap \overline{AB}$. Then points $B,C,E,F$ are concyclic. Using $BPCQ$ is a parallelogram we get
$$ \angle QCB = \angle PBC = \angle EBC = \angle EFC = \angle EFP $$Similarly $\angle QBC = \angle FEP$. Hence,
$$ \triangle QBC \sim \triangle PEF $$[asy]
size(200);
pair B=dir(-160),C=dir(-20),E=dir(70),F=dir(135),A=extension(B,F,C,E),P=extension(B,E,C,F),Q=B+C-P;
draw(unitcircle,cyan);
fill(P--E--F--P--cycle,purple+grey+grey);
fill(B--Q--C--B--cycle,purple+grey+grey);
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$E$",E,dir(E));
dot("$F$",F,dir(F));
dot("$P$",P,dir(-90));
dot("$Q$",Q,dir(Q));
draw(A--B--C--A,royalblue);
draw(B--E^^C--F,red);
draw(P--A--Q,green);
draw(B--Q--C^^E--F,brown);
[/asy]
Let $\mathbb H$ denote homothety at $A$ with scale $\frac{AF}{AC} = \frac{AE}{AB}$ followed by reflection in internal angle bisector of $\angle BAC$. Note $\mathbb H(F) = C$ and $\mathbb H(E) = B$. Thus $\mathbb H(P) = P'$ is a point such that $$\triangle PEF \sim \triangle P' BC$$Hence $P' \equiv Q$. As $\mathbb H$ also consists of reflection in internal angle bisector of $\angle BAC$, so $\angle BAP = \angle CAQ$ follows. $\blacksquare$
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TechnoLenzer
55 posts
#13 • 2 Y
Y by ike.chen, allin27x
Let $\infty_1 = BP \cap CQ$ and $\infty_2 = CP \cap BQ$. Since these are parallel pairs of lines, $\infty_1, \infty_2$ are the points at infinity for those pencils of parallel lines respectively. Note that $\measuredangle \infty_1AB = \measuredangle PBA = \measuredangle ACP = \measuredangle CA\infty_2$ by $A\infty_1 \; || \; CP$ and $A\infty_2 \; || \; BP$. Thus, $A\infty_1$ is isogonal to $A\infty_2$ wrt. $\triangle ABC$. Hence by DDIT on complete quadrilateral $P, B, Q, C, \infty_1, \infty_2$, there exists a projective involution swapping $(AB, AC)$, $(A\infty_1, A\infty_2)$, $(AP, AQ)$. This is taking the isogonal, and so $AP, AQ$ are isogonal. $\blacksquare$
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samrocksnature
8791 posts
#14
Y by
Any complex sols?
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AwesomeYRY
579 posts
#15
Y by
Consider the following two series of laws of sines:
\begin{align*}
    \frac{BQ}{\sin(\angle BAQ)} = \frac{AQ}{\sin(\angle ABP + \angle PBQ)} &= \frac{AQ}{\sin(\angle PCA + \angle ACP)} = \frac{CQ}{\sin (\angle QAC)},\\
    \frac{BP}{\sin(\angle BAP)} = \frac{AP}{\sin(\angle ABP)} &= \frac{AP}{\sin(\angle PCA)} = \frac{PC}{\sin(\angle PAC)}.
\end{align*}Putting them together, we get
\[\frac{\sin(\angle QAC)}{\sin(\angle BAQ)} = \frac{CQ}{BQ}= \frac{BP}{PC} = \frac{\sin(\angle BAP)}{\sin(\angle PAC)}\]and since $\angle WAC + \angle BAQ = \angle BAC = \angle BAP + \angle PAC$, we have that $\angle BAP = \angle QAC$ and we're done. $\blacksquare$.
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anurag27826
93 posts
#16
Y by
Amazing problem, although my solution is the same as of guptaamitu1's solution, but still im posting it for the sake of storage.

First of all we claim that $\triangle PEF \sim \triangle BQC$. Note that $\angle EPF = \angle BPC = \angle BQC$. Also note that $\angle PFE = \angle PBC = \angle QCB$. Both angle equalities are there since $BPCQ$ is a parallelogram. So, consider homothety $\psi$ under $A$ follow by reflection along the angle bisector of $\angle BAC$ with scale $\frac{AF}{AC}$. Note that $\psi$ sends $F$ to $C$ and $E$ to $B$. Then $\psi$ sends $P$ to $P'$ such that $\triangle CP'B \sim FPE \implies P' = Q$. So, it also implies that the line $AQ$ is a reflection of the line $AP$ along the angle bisector of $\angle BAC$, which implies $\angle BAP = \angle CAQ$. So, we're done.
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OronSH
1748 posts
#17
Y by
Great problem.

We use directed angles. Let $P'$ be the isogonal conjugate of $P.$ Consider the lines through $P'$ parallel to $PB,PC.$ These lines intersect $AB$ at points $D,F$ with $D$ between $A$ and $F,$ and they intersect $AC$ at points $E,G$ with $E$ between $A$ and $G.$ Since $\angle DFE=\angle ABP=\angle PCA=\angle DGE,$ we have that $DEGF$ is cyclic. Also, since $\angle BFP'=\angle ABP=\angle PCA=\angle BCP',$ we have $BFCP'$ is cyclic, and similarly $BCGP'$ is cyclic, so $BFCG$ is cyclic, and by Reim's theorem we have $DE,BC$ parallel. Now, take a homothety at $A$ sending $DE$ to $BC.$ It is not hard to see that this takes $P'$ to $Q,$ so $A,P',Q$ are collinear, and we have $\angle QAB=\angle CAP.$
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huashiliao2020
1292 posts
#18
Y by
It states that for a point P inside a triangle ABC with ABP=ACP, the point Q such that BPCQ is a parallelogram, AQ-AP are pairwise isogonal lines.

Here's the first one, quoted from my sl2012g2 (a good application for anyone who's reading!)
Sketch. If we let D be the point s.t. APBD is a parallelogram, we have that BDQ is congruent to APC by a translation of vector AD. (This is just done by length equalities and parallel lines.) Now, from DQB=ACP=ABP=DAB, ADBQ is cyclic. Then BAQ=BRQ=PAC, as desired. $\square$

Second solution I just came up with: Let BP,CP intersect AC,AB at E,F, respectively. It's obvious that $$F\in(BCE)\implies EFP=EBC=QCB,FEP=FCB=CBQ\implies EFP\sim CBQ,AEF\sim ABC\implies AEPF\sim ABQC\implies BAP=FAP=CAQ,$$as desired. $\blacksquare$
This post has been edited 2 times. Last edited by huashiliao2020, Aug 30, 2023, 11:01 PM
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Ianis
416 posts
#19
Y by
Let $P'$ be the isogonal conjugate of $P$ in $ABC$, the angle condition implies that $P'$ is on the perpendicular bisector of $BC$. Use barycentric coordinates with respect to $ABC$ and let $P=(x,y,z)$. Then $P'=\left (\frac{a^2}{x}:\frac{b^2}{y}:\frac{c^2}{z}\right )$ and $Q=B+C-P=(-x,1-y,1-z)=(-x,z+x,x+y)$. We have\begin{align*}\begin{vmatrix}1&0&0\\\dfrac{a^2}{x}&\dfrac{b^2}{y}&\dfrac{c^2}{z}\\-x&z+x&x+y\end{vmatrix} & =\begin{vmatrix}\dfrac{b^2}{y}&\dfrac{c^2}{z}\\z+x&x+y\end{vmatrix} \\
& =b^2\frac{x+y}{y}-c^2\frac{z+x}{z} \\
& =\frac{b^2yz-c^2yz+b^2zx-c^2xy}{yz} \\
& =\frac{x}{a^2}\left ((b^2-c^2)\frac{a^2}{x}+a^2\left (\frac{b^2}{y}-\frac{c^2}{z}\right )\right ) \\
& =0,
\end{align*}where the last equality holds because $P'$ is on the perpendicular bisector of $BC$. Hence $A,P',Q$ are collinear, so $\angle QAB=\angle P'AB=\angle CAP$. Done.
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asdf334
7585 posts
#20
Y by
hello;;; ddit with $A$ and $BPCQ$
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shendrew7
799 posts
#21
Y by
Trivial by first isogonality lemma.

Let $X = BP \cap CA$ and $Y = CP \cap AB$. Then $BCXY$ is cyclic and $BPCQ$ is a parallelogram, so
\[\angle PXY = \angle BCP = \angle CBQ, \quad \angle PXY = \angle CBP = \angle BCQ.\]
Hence we have $\triangle PXY \sim \triangle QBC$ as well as $\triangle AXY \sim \triangle ABC$, so the quadrilaterals $AXPY$ and $ABQC$ are also similar with opposite orientation, which implies the desired. $\blacksquare$
This post has been edited 1 time. Last edited by shendrew7, Dec 31, 2023, 7:14 AM
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joshualiu315
2534 posts
#22
Y by
Let $R$ be the reflection of $P$ across the midpoint of $\overline{AB}$. It is clear that $ARBP$ is a parallelogram. Then, notice that $\overline{AR} \parallel \overline{BP} \parallel \overline{QC}$, and $AR=BP=QC$. Hence, $ARQC$ is a parallelogram, meaning that $\angle ARQ = \angle ACQ$.

Also, we have

\[\angle ABQ = \angle PBQ+\angle ABP =\angle PCQ+\angle PCA = \angle ACQ.\]
This means that $\angle ABQ = \angle ARQ$, so $ARBQ$ is cyclic. Thus,

\[\angle BAP = \angle ABR = \angle AQR = \angle CAQ. \ \square\]
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EpicBird08
1755 posts
#23
Y by
Let $H$ be the point such that $APCH$ is a parallelogram. First note that $AH = CP = BQ$ since $BPCQ$ is a parallelogram as well. Additionally, this gives $AH \parallel PC \parallel BQ,$ so $AHQB$ is also a parallelogram. Then $$\measuredangle AHQ = \measuredangle QBA = \measuredangle QBP + \measuredangle PBA = \measuredangle PCQ + \measuredangle ACP = \measuredangle ACQ,$$so $AHCQ$ is cyclic.

Next, note from our three parallelograms that $QC = BP, CH = AP,$ and $QH = AB,$ so $\triangle ABP \cong \triangle HQC.$ Finally, $\angle BAP = \angle QHC = \angle QAC,$ as desired.
This post has been edited 1 time. Last edited by EpicBird08, Jan 13, 2024, 5:09 PM
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dolphinday
1328 posts
#24
Y by
Construct parallelograms $APCR$, $APBS$, and $BPCQ$.
And it follows that $ARQB$ is a parallelogram since $BQ \parallel PC \parallel AR$ and $AP \parallel RC$. Similarly, $ACQS$ is a parallelogram. Then we have $\angle SBP = 180^{\circ} = \angle SAP = \angle QAR = \angle BQA$ so $ASBQ$ is cyclic. Then $\angle CAQ = \angle{AQS} = \angle SBA = \angle{PAB}$ so we are done.
This post has been edited 2 times. Last edited by dolphinday, Feb 4, 2024, 3:53 PM
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Assassino9931
1364 posts
#25
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Denote $\angle BAC = \alpha$, $\angle ABP = \angle ACP = x$, $\angle BAP = y$, $\angle CAQ = z$. The Sine Law in $ABP$ and $ACP$ gives $\frac{BP}{\sin y} = \frac{AP}{\sin x} = \frac{CP}{\sin(\alpha - y)}$, i.e. $\frac{CP}{BP} = \frac{\sin(\alpha - y)}{\sin y}$. Analogously $\frac{BQ}{CQ} = \frac{\sin(\alpha - z)}{\sin z}$. However, $CP = BQ$ and $BP = CQ$ from the parallelogram $BPCQ$, thus $\frac{\sin(\alpha - y)}{\sin y} = \frac{\sin(\alpha - z)}{\sin z}$. Hence $\sin\alpha \cot y - \cos \alpha = \sin\alpha \cot z - \cos \alpha$, equivalent to $\cot y = \cot z$, i.e. $y=z$, as desired.
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dolphinday
1328 posts
#26
Y by
Alternate solution involving DDIT;

Apply DDIT on quadrilateral $BC\infty_{BP}\infty_{BQ}$ from point $A$.
Then $B\infty_{BP} \cap C\infty_{BQ} = P$, and $B_\infty{BQ} \cap C\infty_{BP} = Q$, so we get that $(AB, AC)$, $(A\infty_{BP}, A\infty_{BQ})$, and $(AP, AQ)$ are involutions.
However $\angle \infty_{BQ}AC = \angle ACP = \angle ABP = \angle \infty_{BP}AB$, so $(AB, AC)$ and $(A\infty_{BP}, A\infty_{BQ})$ are both involutions around the angle bisector of $\angle BAC$. So it follows that $AP$ and $AQ$ are isogonal as desired.
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Martin2001
164 posts
#27
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Let the reflection of $P$ over the midpoint of $AC$ be $K.$ We see that $AKQB$ is a parralelogram. We show $AKCQ$ is cyclic because then $\angle QAC=\angle QKC=\angle BAQ.$ Let $\angle QAC=y, \angle ACP=x.$ Then $a-y=\angle BAQ=\angle KQA.$ Then note that $\angle CAK=x.$ Therefore $$180-a-x=\angle AKQ=\angle ACQ=b+c-x,$$as desired$.\blacksquare$
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ihatemath123
3449 posts
#28 • 2 Y
Y by OronSH, happypi31415
Vertically stretch about the angle bisector of $\angle BAC$ until $P$ is the orthocenter of $\triangle ABC$, then $Q$ is the antipode so it's obvious.
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bachkieu
137 posts
#29
Y by
I think this works?
Let $CQ \cap AB = X, BQ \cap AC = Y$; it's easy to show that $\triangle ABC \sim \triangle AYX$ and that $P$ of $\triangle$ $ABC$ corresponds to $Q$ of $\triangle AYX$.
This post has been edited 1 time. Last edited by bachkieu, Sep 5, 2024, 12:14 AM
Reason: forgot period
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Lleeya
11 posts
#30 • 1 Y
Y by endless_abyss
This is Romanian Lemma, construct $EPFR$ to be parralelogram and trivial by similarity of $AEF$ and $ABC$.
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endless_abyss
52 posts
#31
Y by
Here's my solution :)
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This post has been edited 1 time. Last edited by endless_abyss, Nov 25, 2024, 5:31 PM
Reason: typo haha
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AndreiVila
209 posts
#32 • 2 Y
Y by trigadd123, Ciobi_
Lleeya wrote:
This is Romanian Lemma, construct $EPFR$ to be parralelogram and trivial by similarity of $AEF$ and $ABC$.

Now I can finally go to sleep knowing that we've achieved our goal at MOP. We won.
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AshAuktober
1009 posts
#33
Y by
Yay!
The given is equivalent to $A\infty_{BP}$, $A\infty_{CP}$ being isogonal. From here Isogonal Line Lemma (or DDIT if you wish) gives the required.
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Tsikaloudakis
981 posts
#35
Y by
see the figure:
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zuat.e
66 posts
#36
Y by
Let $E,F=CP\cap AB, BP\cap AC$, then $EFBC$ is cyclic and $\triangle PEF\sim \triangle QCB$.
Now, we consider the transformation composed by the homothety of ratio $\frac{AC}{AF}$ and its reflection $w.r.t.$ the $\angle A$ angle bisector.
Clearly $E$ is sent to $B$ and $F$ to $C$ and as this transformation preserves similarity, it sends $P$ to $Q$, hence $AP$ and $AQ$ are isogonal.
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happypi31415
754 posts
#37 • 1 Y
Y by dolphinday
I think this might be a fakesolve because its way dumber then all the other solutions here :/

edit: never mind its the same as shendrew's solution. the solution with constructing parallelograms is really cool :o

Extend $BP$ and $CP$ to intersect $AC$ and $AB$ at $X,Y$ respectively. Then, $BCXY$ is cyclic and thus we have $\triangle AXY \sim \triangle ABC$. Furthermore, note that $\angle XYP = \angle XBC = \angle QCB$ (with similar logic following for $\angle YXP$ and $\angle CBQ$) so in fact, $AYPX$ and $ABQC$ are homothetic to each other. Then, since they are oppositely oriented but share $\angle BAC$, it thus follows that they must be reflections about the angle bisector and thus $\angle BAP = \angle CAQ$ as desired.
This post has been edited 2 times. Last edited by happypi31415, May 13, 2025, 12:57 AM
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