We have your learning goals covered with Spring and Summer courses available. Enroll today!

G
Topic
First Poster
Last Poster
k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Sunday, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
Sunday, Apr 13 - Aug 10
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Tuesday, Mar 25 - Jul 8
Sunday, Apr 13 - Aug 10
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21


Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, Mar 23 - Jul 20
Monday, Apr 7 - Jul 28
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Sunday, Mar 16 - Jun 8
Wednesday, Apr 16 - Jul 2
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Monday, Mar 17 - Jun 9
Thursday, Apr 17 - Jul 3
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Sunday, Mar 2 - Jun 22
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
Wednesday, Apr 23 - Oct 1
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
Monday, Apr 21 - Oct 13
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Sunday, Mar 23 - Aug 3
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Friday, Apr 11 - Jun 27
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Sunday, Mar 16 - Aug 24
Wednesday, Apr 9 - Sep 3
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Wednesday, Mar 5 - May 21
Tuesday, Jun 10 - Aug 26

Calculus
Sunday, Mar 30 - Oct 5
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Sunday, Mar 23 - Jun 15
Wednesday, Apr 16 - Jul 2
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

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
Monday, Mar 24 - Jun 16
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Sunday, Mar 30 - Jun 22
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Tuesday, Mar 25 - Sep 2
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Mar 2, 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
Polynomial produces perfect powers
TheUltimate123   21
N 3 minutes ago by pi271828
Source: ELMO 2023/1
Let \(m\) be a positive integer. Find, in terms of \(m\), all polynomials \(P(x)\) with integer coefficients such that for every integer \(n\), there exists an integer \(k\) such that \(P(k)=n^m\).

Proposed by Raymond Feng
21 replies
TheUltimate123
Jun 26, 2023
pi271828
3 minutes ago
geometry
srnjbr   0
5 minutes ago
the points f,n,o, t a lie in the plane such that the triangles tfo ton are similar, preserving direction and order, and fano is a parallelogram. show that of×on=oa×ot.
0 replies
srnjbr
5 minutes ago
0 replies
Problem 5
blug   0
22 minutes ago
Source: Polish Junior Math Olympiad Finals 2025
Each square on a 5×5 board contains an arrow pointing up, down, left, or right. Show that it is possible to remove exactly 20 arrows from this board so that no two of the remaining five arrows point to the same square.
0 replies
blug
22 minutes ago
0 replies
Diophantine equation with large moduli
Assassino9931   2
N 23 minutes ago by Assassino9931
Source: Bulgaria, Concours Generale Minko Balkanski 2024
Solve in positive integers $2^x - 23^y = 9$.
2 replies
Assassino9931
4 hours ago
Assassino9931
23 minutes ago
Problem 4
blug   0
24 minutes ago
Source: Polish Junior Math Olympiad Finals 2025
In a rhombus $ABCD$, angle $\angle ABC=100^{\circ}$. Point $P$ lies on $CD$ such that $\angle PBC=20^{\circ}$. Line parallel to $AD$ passing trough $P$ intersects $AC$ at $Q$. Prove that $BP=AQ$.
0 replies
blug
24 minutes ago
0 replies
Problem 3
blug   0
26 minutes ago
Source: Polish Junior Math Olympiad Finals 2025
Find all primes $(p, q, r)$ such that
$$pq+4=r^4.$$
0 replies
blug
26 minutes ago
0 replies
Problem 2
blug   0
27 minutes ago
Source: Polish Junior Math Olympiad Finals 2025
A party is attended by boys and girls. Each person attending the party knows exactly 3 boys and exactly 7 girls among the other people. Prove that the number of all the people attending the party is divisible by 20.
0 replies
blug
27 minutes ago
0 replies
Problem 1
blug   0
28 minutes ago
Source: Polish Junior Math Olympiad Finals 2025
Do there exists a tetrahedron, in which the lenghts of the edges are six different integers such that their sum is 25?
0 replies
blug
28 minutes ago
0 replies
A two-variable & non-homogenous inequality that seems hard to me
MyLifeMyChoice   3
N 36 minutes ago by Radin_
Source: Developing from a larger, three-variable one
For $a,b>0$, prove/disprove the following claim: :maybe:

$a^3b^3+\frac{1}{a^3}+\frac{1}{b^3}+3\stackrel{?}{\ge}a^2b+b^2a+\frac{1}{a^2b}+\frac{1}{b^2a}+\frac{a}{b}+\frac{b}{a}$
3 replies
MyLifeMyChoice
Mar 13, 2025
Radin_
36 minutes ago
exponential diophantine with factorials
skellyrah   4
N an hour ago by InftyByond
find all non negative integers (x,y) such that $$ x! + y! = 2025^x + xy$$
4 replies
skellyrah
Feb 24, 2025
InftyByond
an hour ago
Point satisfies triple property
62861   35
N an hour ago by Sanjana42
Source: USA Winter Team Selection Test #2 for IMO 2018, Problem 2
Let $ABCD$ be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at $H$. Denote by $M$ and $N$ the midpoints of $\overline{BC}$ and $\overline{CD}$. Rays $MH$ and $NH$ meet $\overline{AD}$ and $\overline{AB}$ at $S$ and $T$, respectively. Prove that there exists a point $E$, lying outside quadrilateral $ABCD$, such that
[list]
[*] ray $EH$ bisects both angles $\angle BES$, $\angle TED$, and
[*] $\angle BEN = \angle MED$.
[/list]

Proposed by Evan Chen
35 replies
62861
Jan 22, 2018
Sanjana42
an hour ago
Prove concyclic and tangency
syk0526   40
N an hour ago by Ilikeminecraft
Source: Japan Olympiad Finals 2014, #4
Let $ \Gamma $ be the circumcircle of triangle $ABC$, and let $l$ be the tangent line of $\Gamma $ passing $A$. Let $ D, E $ be the points each on side $AB, AC$ such that $ BD : DA= AE : EC $. Line $ DE $ meets $\Gamma $ at points $ F, G $. The line parallel to $AC$ passing $ D $ meets $l$ at $H$, the line parallel to $AB$ passing $E$ meets $l$ at $I$. Prove that there exists a circle passing four points $ F, G, H, I $ and tangent to line $ BC$.
40 replies
syk0526
May 17, 2014
Ilikeminecraft
an hour ago
p^2+3*p*q+q^2
mathbetter   0
2 hours ago
\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]
0 replies
+1 w
mathbetter
2 hours ago
0 replies
two sequences of positive integers and inequalities
rmtf1111   49
N 2 hours ago by dolphinday
Source: EGMO 2019 P5
Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions:

$\text{(A)} \ a_i\le b_i$ for $i=1, 2, \cdots , n;$

$\text{(B)} \ $ the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and

$\text{(C)} \ $ $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$

(Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)
49 replies
rmtf1111
Apr 10, 2019
dolphinday
2 hours ago
triangles with equal areas
mathuz   6
N Jun 29, 2024 by PROF65
Source: SRMC 2023, P1
Let $ABCD$ be a trapezoid with $AD\parallel BC$. A point $M $ is chosen inside the trapezoid, and a point $N$ is chosen inside the triangle $BMC$ such that $AM\parallel CN$, $BM\parallel DN$. Prove that triangles $ABN$ and $CDM$ have equal areas.
6 replies
mathuz
Dec 28, 2023
PROF65
Jun 29, 2024
triangles with equal areas
G H J
G H BBookmark kLocked kLocked NReply
Source: SRMC 2023, P1
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathuz
1510 posts
#1
Y by
Let $ABCD$ be a trapezoid with $AD\parallel BC$. A point $M $ is chosen inside the trapezoid, and a point $N$ is chosen inside the triangle $BMC$ such that $AM\parallel CN$, $BM\parallel DN$. Prove that triangles $ABN$ and $CDM$ have equal areas.
This post has been edited 1 time. Last edited by mathuz, Dec 28, 2023, 7:42 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
SBYT
196 posts
#2
Y by
Nice problem!I think it must can be solved by just area chasing,but I just find this solution which is not beautiful.

Let $AMCK$ and $ABCJ$ are parallelograms,easy to know that $\vartriangle ABM\cong\vartriangle CJK$ and $\vartriangle ABK\cong\vartriangle CJM$.Let $AB$ meets $CN$ at $H$,meets $CM$ at $I$,$AJ$ meets $CM$ at $G$.
$\frac{IH}{IB}=\frac{IH}{IA}\cdot\frac{IA}{IB}=\frac{IC}{IM}\cdot\frac{IG}{IC}=\frac{IG}{IM}$,so $GH\parallel MB$,so $GH\parallel DN\parallel JK$.
$S_\vartriangle ABN=S_\vartriangle ABK\cdot\frac{HN}{HK}=S_\vartriangle CJM\cdot\frac{GD}{GJ}=S_\vartriangle CDM$.$\Box$
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathuz
1510 posts
#3
Y by
I'm not sure of any other good synthetic way. But, the problem can be done using Cartesian coordinate system and the area formula for a triangle.

More precisely, if we assume $A=(a,y_1)$, $B=(b,y_2)$, $C=(c,y_2)$, $D=(d,y_1)$ and $M=(m_1,m_2)$, $N=(n_1,n_2)$ (we are assuming $AD$ is parallel to $x$-axis), then we should have \[ \frac{m_2-y_1}{m_1-a} = \frac{y_2-n_2}{c-n_1} \quad \text{and} \quad \frac{n_2-y_1}{n_1-d} = \frac{y_2-m_2}{b-m_1} \quad \quad (1)  \]as $AM\parallel CN$ and $BM\parallel DN$.
We only need to show that \[ \frac{1}{2} \begin{vmatrix}
1 & 1 & 1\\
a & b & n_1 \\
y_1 & y_2 & n_2
\end{vmatrix} = \frac{1}{2} \begin{vmatrix}
1 & 1 & 1\\
c & d & m_1 \\
y_2 & y_1 & m_2
\end{vmatrix}   \quad \quad (2) \]as they are areas of triangles $ABN$ (left one) and $CDM$ (right one).

It can be shown that $(1)$ implies $(2)$ using some algebraic manipulations.
This post has been edited 1 time. Last edited by mathuz, Jan 1, 2024, 8:34 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hellnish
27 posts
#4
Y by
Does anyone have pure geometry solution?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sami1618
871 posts
#5 • 1 Y
Y by ehuseyinyigit
Define $X=AM\cap DN$ and $Y=BM\cap CN$. Let $\angle MYN=\theta$ and let $\lambda=\frac{1}{2}\sin(\theta)$. $$[ABN]=[AYN]+[NYB]+[BYA]=\lambda(YM\cdot YN+YB\cdot YN+YB\cdot AM)=\lambda(YM\cdot YN+YB\cdot AX)$$Similarly $[CDM]=\lambda(XM\cdot XN+XD\cdot CY)$. We are done as $YM\cdot YN=XM\cdot XN$ and $YB\cdot AX=XD\cdot CY$ (because $\Delta AXD\sim\Delta CYB$).

Remark: An alternative approach would be to send parallelogram $MYNX$ to a unit square through an affine transformation.
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Ianis
394 posts
#6
Y by
Use barycentric coordinates with respect to $ABC$. Then $D=(1,t,-t)$ for some $t\in \mathbb{R}$. Let $M=(x,y,z)$. Then $P_{\infty AM}=(-y-z:y:z)$ and $P_{\infty BM}=(x:-x-z:z)$, so $N=(-y-z:y:s)$ for some $s\in \mathbb{R}$ (because $AM\parallel CN$) and $N\in DP_{\infty BM}$ (because $BM\parallel DN$), hence $N=\left (-y-z:y:z\frac{tx-y}{(t+1)x+z}\right )=(-(y+z)((t+1)x+z):y((t+1)x+z):z(tx-y))$. Then using $x+y+z=1$ we get $N=\left (\frac{(y+z)((t+1)x+z)}{z},-\frac{y((t+1)x+z)}{z},y-tx\right )$. Hence$$\frac{[ABM]}{[ABC]}=\begin{vmatrix}1&0&0\\0&1&0\\\frac{(y+z)((t+1)x+z)}{z}&-\frac{y((t+1)x+z)}{z}&y-tx\end{vmatrix}=y-tx$$and$$\frac{[CDM]}{[ABC]}=\begin{vmatrix}0&0&1\\1&t&-t\\x&y&z\end{vmatrix}=y-tx.$$Therefore, $[ABM]=[CDM]$, as desired.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
PROF65
2016 posts
#7 • 2 Y
Y by sami1618, GeoKing
The result is valid though $M, N$ are outside or inside the specified polygon.
Denotes the signed area of the polygon $X$ as $[X]$
Lemma
Let $ABCD$ be a trapezoid s.t. $CD\parallel AB$ ; $E  $ be a point then
$[AECD]=[BECD]$
the proof is easy.
Applying the lemma twice we get $[BAND]=[CAND ]=[CMND ]$ but $[BAND]=[BAN ]+[BND ]$ and $[CMND]=[DCM ]+[DMN ] $ besides
$[BND ]=[DMN ](\because DN\parallel BM)$ we deduces then $[BAN ]=[DCM ]$
best regards.
RH HAS
Z K Y
N Quick Reply
G
H
=
a