Plan ahead for the next school year. Schedule your class today!

G
Topic
First Poster
Last Poster
k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Wednesday, Jul 16 - Oct 29
Sunday, Aug 17 - Dec 14
Tuesday, Aug 26 - Dec 16
Friday, Sep 5 - Jan 16
Monday, Sep 8 - Jan 12
Tuesday, Sep 16 - Jan 20 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Sep 21 - Jan 25
Thursday, Sep 25 - Jan 29
Wednesday, Oct 22 - Feb 25
Tuesday, Nov 4 - Mar 10
Friday, Dec 12 - Apr 10

Prealgebra 2 Self-Paced

Prealgebra 2
Friday, Jul 25 - Nov 21
Sunday, Aug 17 - Dec 14
Tuesday, Sep 9 - Jan 13
Thursday, Sep 25 - Jan 29
Sunday, Oct 19 - Feb 22
Monday, Oct 27 - Mar 2
Wednesday, Nov 12 - Mar 18

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Tuesday, Jul 15 - Oct 28
Sunday, Aug 17 - Dec 14
Wednesday, Aug 27 - Dec 17
Friday, Sep 5 - Jan 16
Thursday, Sep 11 - Jan 15
Sunday, Sep 28 - Feb 1
Monday, Oct 6 - Feb 9
Tuesday, Oct 21 - Feb 24
Sunday, Nov 9 - Mar 15
Friday, Dec 5 - Apr 3

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Wednesday, Jul 2 - Sep 17
Sunday, Jul 27 - Oct 19
Monday, Aug 11 - Nov 3
Wednesday, Sep 3 - Nov 19
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Friday, Oct 3 - Jan 16
Sunday, Oct 19 - Jan 25
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8

Introduction to Number Theory
Tuesday, Jul 15 - Sep 30
Wednesday, Aug 13 - Oct 29
Friday, Sep 12 - Dec 12
Sunday, Oct 26 - Feb 1
Monday, Dec 1 - Mar 2

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Friday, Jul 18 - Nov 14
Thursday, Aug 7 - Nov 20
Monday, Aug 18 - Dec 15
Sunday, Sep 7 - Jan 11
Thursday, Sep 11 - Jan 15
Wednesday, Sep 24 - Jan 28
Sunday, Oct 26 - Mar 1
Tuesday, Nov 4 - Mar 10
Monday, Dec 1 - Mar 30

Introduction to Geometry
Monday, Jul 14 - Jan 19
Wednesday, Aug 13 - Feb 11
Tuesday, Aug 26 - Feb 24
Sunday, Sep 7 - Mar 8
Thursday, Sep 11 - Mar 12
Wednesday, Sep 24 - Mar 25
Sunday, Oct 26 - Apr 26
Monday, Nov 3 - May 4
Friday, Dec 5 - May 29

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22
Friday, Aug 8 - Feb 20
Tuesday, Aug 26 - Feb 24
Sunday, Sep 28 - Mar 29
Wednesday, Oct 8 - Mar 8
Sunday, Nov 16 - May 17
Thursday, Dec 11 - Jun 4

Intermediate Counting & Probability
Sunday, Sep 28 - Feb 15
Tuesday, Nov 4 - Mar 24

Intermediate Number Theory
Wednesday, Sep 24 - Dec 17

Precalculus
Wednesday, Aug 6 - Jan 21
Tuesday, Sep 9 - Feb 24
Sunday, Sep 21 - Mar 8
Monday, Oct 20 - Apr 6
Sunday, Dec 14 - May 31

Advanced: Grades 9-12

Calculus
Sunday, Sep 7 - Mar 15
Wednesday, Sep 24 - Apr 1
Friday, Nov 14 - May 22

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Wednesday, Sep 3 - Nov 19
Tuesday, Sep 16 - Dec 9
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Oct 6 - Jan 12
Thursday, Oct 16 - Jan 22
Tues, Thurs & Sun, Dec 9 - Jan 18 (meets three times a week!)

MATHCOUNTS/AMC 8 Advanced
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Tuesday, Aug 26 - Nov 11
Thursday, Sep 4 - Nov 20
Friday, Sep 12 - Dec 12
Monday, Sep 15 - Dec 8
Sunday, Oct 5 - Jan 11
Tues, Thurs & Sun, Dec 2 - Jan 11 (meets three times a week!)
Mon, Wed & Fri, Dec 8 - Jan 16 (meets three times a week!)

AMC 10 Problem Series
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 10 - Nov 2
Thursday, Aug 14 - Oct 30
Tuesday, Aug 19 - Nov 4
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Mon, Wed & Fri, Oct 6 - Nov 3 (meets three times a week!)
Tue, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 10 Final Fives
Friday, Aug 15 - Sep 12
Sunday, Sep 7 - Sep 28
Tuesday, Sep 9 - Sep 30
Monday, Sep 22 - Oct 13
Sunday, Sep 28 - Oct 19 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, Oct 8 - Oct 29
Thursday, Oct 9 - Oct 30

AMC 12 Problem Series
Wednesday, Aug 6 - Oct 22
Sunday, Aug 10 - Nov 2
Monday, Aug 18 - Nov 10
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
Sunday, Sep 28 - Oct 19
Tuesday, Oct 7 - Oct 28

AIME Problem Series A
Thursday, Oct 23 - Jan 29

AIME Problem Series B
Tuesday, Sep 2 - Nov 18

F=ma Problem Series
Tuesday, Sep 16 - Dec 9
Friday, Oct 17 - Jan 30

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT


Programming

Introduction to Programming with Python
Thursday, Aug 14 - Oct 30
Sunday, Sep 7 - Nov 23
Tuesday, Dec 2 - Mar 3

Intermediate Programming with Python
Friday, Oct 3 - Jan 16

USACO Bronze Problem Series
Wednesday, Sep 3 - Dec 3
Thursday, Oct 30 - Feb 5
Tuesday, Dec 2 - Mar 3

Physics

Introduction to Physics
Tuesday, Sep 2 - Nov 18
Sunday, Oct 5 - Jan 11
Wednesday, Dec 10 - Mar 11

Physics 1: Mechanics
Sunday, Sep 21 - Mar 22
Sunday, Oct 26 - Apr 26
0 replies
jwelsh
Jul 1, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
AOPS MO Introduce
MathMaxGreat   83
N a minute ago by Ciobi_
$AOPS MO$

Problems: post it as a private message to me or @jerryZYang, please post it in $LATEX$ and have answers

6 Problems for two rounds, easier than $IMO$

If you want to do the problems or be interested, reply ’+1’
Want to post a problem reply’+2’ and message me
Want to be in the problem selection committee, reply’+3’
83 replies
+1 w
MathMaxGreat
Jul 12, 2025
Ciobi_
a minute ago
Problem 4 of the HMO
GreekIdiot   13
N 25 minutes ago by Diaoest
Prove that no perfect cube is of the form $y^2+108$ where $y \in \mathbb{Z}$.
13 replies
GreekIdiot
Feb 22, 2025
Diaoest
25 minutes ago
Convex quad
MithsApprentice   83
N 31 minutes ago by mathprodigy2011
Source: USAMO 1993
Let $\, ABCD \,$ be a convex quadrilateral such that diagonals $\, AC \,$ and $\, BD \,$ intersect at right angles, and let $\, E \,$ be their intersection. Prove that the reflections of $\, E \,$ across $\, AB, \, BC, \, CD, \, DA \,$ are concyclic.
83 replies
MithsApprentice
Oct 27, 2005
mathprodigy2011
31 minutes ago
Areas of triangles AOH, BOH, COH
Arne   72
N an hour ago by mudkip42
Source: APMO 2004, Problem 2
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Prove that the area of one of the triangles $AOH$, $BOH$ and $COH$ is equal to the sum of the areas of the other two.
72 replies
1 viewing
Arne
Mar 23, 2004
mudkip42
an hour ago
Cauchy-Schwarz proof
Nguyenhuyen_AG   1
N an hour ago by Nguyenhuyen_AG
Let $a, \, b, \, c$ be non-negative real numbers. Prove that
\[\frac{8a}{b+2c}+\frac{48b}{c+3a}+\frac{c}{a+4b} \geqslant 4.\]\[\frac{16a}{b+2c}+\frac{85b}{c+3a}+\frac{2c}{a+4b} \geqslant 8.\]hide
1 reply
Nguyenhuyen_AG
an hour ago
Nguyenhuyen_AG
an hour ago
inequality
SunnyEvan   3
N an hour ago by SunnyEvan
Source: Own
Let $ x \in [\frac{\pi}{2}-1, 1) $, try to prove or disprove that :
$$ \frac{(\sqrt2 cosx -1)^2}{cos2x+tan\frac{\pi}{8}}-\frac{(\sqrt2 sinx -1)^2}{cos2x-tan\frac{\pi}{8}} \geq \frac{1}{2}(\frac{tanx-1}{tanx+1})^2 $$
3 replies
SunnyEvan
Yesterday at 1:24 PM
SunnyEvan
an hour ago
Easy Combinatorial Geometry
EthanWYX2009   0
an hour ago
Source: 2025 February 谜之竞赛-4
Given integer $n\ge 2$. Define set
\[V_n:=\{(a_1,\cdots ,a_n)\in\mathbb R^n\mid a_1,\cdots ,a_n\ge 0\text{ and }a_1+\cdots +a_n=1\}.\]For $\alpha =(a_1,\cdots ,a_n)$, $\beta =(b_1,\cdots ,b_n)\in V_n$, define $d(\alpha ,\beta):=\sum\limits_{i=1}^n|a_i-b_i|.$

Determine the minimum real number $\lambda$, such that for all $v_1,v_2,\cdots ,v_{n+2}\in V_n$, there exists $1\le i<j\le n+2$ such that $d(x_i,x_j)\le\lambda$.

Created by Cheng Jiang
0 replies
EthanWYX2009
an hour ago
0 replies
Arc Midpoints Form Cyclic Quadrilateral
ike.chen   58
N 2 hours ago by YaoAOPS
Source: ISL 2022/G2
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
58 replies
ike.chen
Jul 9, 2023
YaoAOPS
2 hours ago
A geo shortlist is not a shortlist without a config
Assassino9931   9
N 2 hours ago by ihategeo_1969
Source: RMM Extralist 2021 G4
Let $ABC$ be an acute triangle, let $H$ and $O$ be its orthocentre and circumcentre, respectively,
and let $S$ and $T$ be the feet of the altitudes from $B$ to $AC$ and from $C$ to $AB$, respectively.
Let $M$ be the midpoint of the segment $ST$, and let $N$ be the midpoint of the segment $AH$. The line
through $O$, parallel to $BC$, crosses the sides $AC$ and $AB$ at $F$ and $G$, respectively. The line $NG$
meets the circle $BGO$ again at $K$, and the line $NF$ meets the circle $CFO$ again at $L$. Prove that
the triangles $BCM$ and $KLN$ are similar.
9 replies
Assassino9931
Sep 18, 2023
ihategeo_1969
2 hours ago
AD=BE implies ABC right
v_Enhance   119
N 2 hours ago by mudkip42
Source: European Girl's MO 2013, Problem 1
The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.
119 replies
v_Enhance
Apr 10, 2013
mudkip42
2 hours ago
Inequality
SunnyEvan   3
N 2 hours ago by SunnyEvan
Source: Own
Let $ a,b,c>0 $, try to prove or disprove that: $$ \sum_{\text{cyc}} \frac{14}{39a^4b^2 + 12b^4c^2 + 12c^4a^2} \leq \frac{1}{a^4b^2 + b^4c^2 + c^4a^2} + \sum_{\text{cyc}} \frac{1}{3(b^4c^2 + c^4a^2 + ab^2c^3 )} $$
3 replies
SunnyEvan
Jul 8, 2025
SunnyEvan
2 hours ago
Nice Quadrilateral Geo
amuthup   54
N 2 hours ago by YaoAOPS
Source: 2021 ISL G4
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
54 replies
1 viewing
amuthup
Jul 12, 2022
YaoAOPS
2 hours ago
same vibes
X.Allaberdiyev   45
N 2 hours ago by Lunatic_Lunar7986
Source: JBMO 2025 Problem 3
Let $ABC$ be a right-angled triangle with $\angle A = 90º$, let $D$ be the foot of the altitude from $A$ to $BC$, and let $E$ be the midpoint of $DC$. The circumcircle of $ABD$ intersects $AE$ again at point $F$. Let $X$ be the intersection of the lines $AB$ and $DF$. Prove that $XD = XC$.

Proposed by Dren Neziri, Albania
45 replies
X.Allaberdiyev
Jun 26, 2025
Lunatic_Lunar7986
2 hours ago
Gergonne point Harmonic quadrilateral
niwobin   5
N 3 hours ago by hectorleo123
Triangle ABC has incircle touching the sides at D, E, F as shown.
AD, BE, CF concurrent at Gergonne point G.
BG and CG cuts the incircle at X and Y, respectively.
AG cuts the incircle at K.
Prove: K, X, D, Y form a harmonic quadrilateral. (KX/KY = DX/DY)
5 replies
niwobin
May 17, 2025
hectorleo123
3 hours ago
Fixed and variable points
BR1F1SZ   2
N Jun 4, 2025 by hukilau17
Source: 2025 Francophone MO Seniors P3
Let $\omega$ be a circle with center $O$. Let $B$ and $C$ be two fixed points on the circle $\omega$ and let $A$ be a variable point on $\omega$. We denote by $X$ the intersection point of lines $OB$ and $AC$, assuming $X \neq O$. Let $\gamma$ be the circumcircle of triangle $\triangle AOX$. Let $Y$ be the second intersection point of $\gamma$ with $\omega$. The tangent to $\gamma$ at $Y$ intersects $\omega$ at $I$. The line $OI$ intersects $\omega$ at $J$. The perpendicular bisector of segment $OY$ intersects line $YI$ at $T$, and line $AJ$ intersects $\gamma$ at $P$. We denote by $Z$ the second intersection point of the circumcircle of triangle $\triangle PYT$ with $\omega$. Prove that, as point $A$ varies, points $Y$ and $Z$ remain fixed.
2 replies
BR1F1SZ
May 11, 2025
hukilau17
Jun 4, 2025
Fixed and variable points
G H J
G H BBookmark kLocked kLocked NReply
Source: 2025 Francophone MO Seniors P3
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
BR1F1SZ
590 posts
#1
Y by
Let $\omega$ be a circle with center $O$. Let $B$ and $C$ be two fixed points on the circle $\omega$ and let $A$ be a variable point on $\omega$. We denote by $X$ the intersection point of lines $OB$ and $AC$, assuming $X \neq O$. Let $\gamma$ be the circumcircle of triangle $\triangle AOX$. Let $Y$ be the second intersection point of $\gamma$ with $\omega$. The tangent to $\gamma$ at $Y$ intersects $\omega$ at $I$. The line $OI$ intersects $\omega$ at $J$. The perpendicular bisector of segment $OY$ intersects line $YI$ at $T$, and line $AJ$ intersects $\gamma$ at $P$. We denote by $Z$ the second intersection point of the circumcircle of triangle $\triangle PYT$ with $\omega$. Prove that, as point $A$ varies, points $Y$ and $Z$ remain fixed.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Lam040208
15 posts
#2
Y by
BR1F1SZ wrote:
Let $\omega$ be a circle with center $O$. Let $B$ and $C$ be two fixed points on the circle $\omega$ and let $A$ be a variable point on $\omega$. We denote by $X$ the intersection point of lines $OB$ and $AC$, assuming $X \neq O$. Let $\gamma$ be the circumcircle of triangle $\triangle AOX$. Let $Y$ be the second intersection point of $\gamma$ with $\omega$. The tangent to $\gamma$ at $Y$ intersects $\omega$ at $I$. The line $OI$ intersects $\omega$ at $J$. The perpendicular bisector of segment $OY$ intersects line $YI$ at $T$, and line $AJ$ intersects $\gamma$ at $P$. We denote by $Z$ the second intersection point of the circumcircle of triangle $\triangle PYT$ with $\omega$. Prove that, as point $A$ varies, points $Y$ and $Z$ remain fixed.
Click to reveal hidden text
Attachments:
This post has been edited 1 time. Last edited by Lam040208, Jun 4, 2025, 4:04 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hukilau17
294 posts
#3
Y by
Quick complex bash. Identify $\omega$ with the unit circle so that
$$o=0$$$$|a|=|b|=|c|=1$$$$x=\frac{ac(b-b) - b(-b)(a+c)}{ac-b(-b)} = \frac{b^2(a+c)}{b^2+ac}$$Let $G$ be the center of $\gamma$ so that
$$g = \frac{ax(\overline{a}-\overline{x})}{\overline{a}x-a\overline{x}} = \frac{ax\left(\frac1a-\frac{x}{b^2}\right)}{\frac{x}a-\frac{ax}{b^2}} = \frac{a(ax-b^2)}{a^2-b^2} = \frac{\frac{a^2b^2(a+c)}{b^2+ac}-ab^2}{a^2-b^2} = \frac{ab^2}{b^2+ac}$$Then
$$y = \frac{\overline{a}g}{\overline{g}} = \frac{\frac{b^2}{b^2+ac}}{\frac{c}{b^2+ac}} = \frac{b^2}c$$Note that the coordinate of $Y$ does not depend on $a$. We also have (letting $k$ denote the coordinate of $I$)
$$k = -\frac{y-g}{y\overline{g}-1} = -\frac{\frac{b^2}c-\frac{ab^2}{b^2+ac}}{\frac{b^2}{b^2+ac}-1} = \frac{b^4}{ac^2}$$$$j = -k = -\frac{b^4}{ac^2}$$$$p = j + g - aj\overline{g} = -\frac{b^4}{ac^2} + \frac{ab^2}{b^2+ac} + \frac{b^4}{c(b^2+ac)} = \frac{b^2(a^2c^2-b^4)}{ac^2(b^2+ac)} = \frac{b^2(ac-b^2)}{ac^2}$$Now we find the coordinate of $T$. Since $T$ lies on the perpendicular bisector of $OY$, we have
$$|t-o| = |t-y|$$$$t\overline{t} = \left(t-\frac{b^2}c\right)\left(\overline{t}-\frac{c}{b^2}\right) = t\overline{t} - \frac{ct}{b^2} - \frac{b^2\overline{t}}c + 1$$$$b^4\overline{t} + c^2t = b^2c \implies \overline{t} = \frac{c(b^2-ct)}{b^4}$$Since $T$ lies on line $YI$, we have
$$\overline{t} = \frac{y+k-t}{yk} = \frac{\frac{b^2}c+\frac{b^4}{ac^2}-t}{\frac{b^6}{ac^3}} = \frac{c(ab^2c+b^4-ac^2t)}{b^6}$$Intersecting these gives
$$\frac{c(b^2-ct)}{b^4} = \frac{c(ab^2c+b^4-ac^2t)}{b^6}$$$$b^4-b^2ct = ab^2c+b^4-ac^2t$$$$t = \frac{ab^2}{ac-b^2}$$Let $U$ denote the circumcenter of $\triangle PYT$. We have the vectors
$$p' = p-y = -\frac{b^4}{ac^2}$$$$t' = t-y = \frac{b^4}{c(ac-b^2)}$$Then if $u' = u-y$, we have
\begin{align*}
u' &= \frac{p't'(\overline{p'} - \overline{t'})}{\overline{p'}t' - p'\overline{t'}} \\
&= \frac{-\frac{b^8}{ac^3(ac-b^2)}\left[-\frac{ac^2}{b^4} + \frac{ac^2}{b^2(ac-b^2)}\right]}{-\frac{ac}{ac-b^2} - \frac{b^2}{ac-b^2}} \\
&= \frac{\frac{b^6}{c(ac-b^2)^2} - \frac{b^4}{c(ac-b^2)}}{\frac{ac}{ac-b^2} + \frac{b^2}{ac-b^2}} \\
&= \frac{b^6 - b^4(ac-b^2)}{c(ac-b^2)(ac+b^2)} \\
&= \frac{b^4(2b^2-ac)}{c(a^2c^2-b^4)}
\end{align*}and then
$$u = u' + y = \frac{b^2(a^2c^2-ab^2c+b^4)}{c(a^2c^2-b^4)}$$$$z = \frac{\overline{y}u}{\overline{u}} = \frac{\frac{a^2c^2-ab^2c+b^4}{a^2c^2-b^4}}{\frac{c(b^4-ab^2c+a^2c^2)}{b^2(b^4-a^2c^2)}} = -\frac{b^2}c$$So the coordinate of $Z$ does not depend on $a$ either. $\blacksquare$
Z K Y
N Quick Reply
G
H
=
a