AB^2+AC^2=2BC^2-Prove

Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/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, 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
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
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
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
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
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
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
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
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
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
Tuesday, Jun 10 - Aug 26

Calculus
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
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
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
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
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
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

Apr 2, 2025

0 replies

Cevian cutting triangles, side to perimeter proportional

awesomeming327. 2

N a few seconds ago by starchan

Source: own

For any three points $X$ , $Y$ , $Z$ define $s(XYZ)$ to be the semiperimeter of $\triangle XYZ$ . Let $\triangle ABC$ be a triangle and let $D$ be on side $BC$ such that
$\frac{s(ABD)}{BD}=\frac{s(ACD)}{CD}$ Let $P$ be a point on $AD$ . Let $Q$ and $R$ be on $AB$ and $AC$ such that $AP+AQ=s(ABD)$ and $AP+AR=s(ACD)$ . Prove that there exists a line $\ell$ parallel to $BC$ such that the circumcircles of $APQ$ and $APR$ intersect $\ell$ at two fixed points.

2 replies

+1 w

awesomeming327.

Apr 26, 2025

starchan

a few seconds ago

Involved conditional geo

Assassino9931 3

N 6 minutes ago by Tamam

Source: Balkan MO 2024 Shortlist G4

Let $ABC$ be an acute-angled triangle with $AB < AC$ , orthocenter $H$ , circumcircle $\Gamma$ and circumcentre $O$ . Let $M$ be the midpoint of $BC$ and let $D$ be a point such that $ADOH$ is a parallellogram. Suppose that there exists a point $X$ on $\Gamma$ and on the opposite side of $DH$ to $A$ such that $\angle DXH + \angle DHA = 90^{\circ}$ . Let $Y$ be the midpoint of $OX$ . Prove that if $MY = OA$ , then $OA = 2OH$ .

3 replies

Assassino9931

Yesterday at 10:27 PM

Tamam

6 minutes ago

Iran second round 2025-q1

mohsen 6

N 17 minutes ago by Marco22

Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

6 replies

mohsen

Apr 19, 2025

Marco22

17 minutes ago

function

CarlFriedrichGauss-1777 3

N 32 minutes ago by Blackbeam999

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that:
$f(2021+xf(y))=yf(x+y+2021)$

3 replies

CarlFriedrichGauss-1777

Jun 4, 2021

Blackbeam999

32 minutes ago

No more topics!

AB^2+AC^2=2BC^2-Prove

Pirkuliyev Rovsen 1

N Aug 15, 2014 by professordad

Let the tangents to the circumcircle of a triangle $ABC$ at vertices $B$ and $C$ intersect at point $D$ and let $E$ be the point of intersection of $AD$ and $BC$ .Prove that $AE=ED$ if and only if $AB^2+AC^2=2BC^2$ .

1 reply

Pirkuliyev Rovsen

Aug 15, 2014

professordad

Aug 15, 2014

AB^2+AC^2=2BC^2-Prove

G H J

Reveal topic tags

G H BBookmark kLocked kLocked NReply

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

Pirkuliyev Rovsen

5047 posts

Pirkuliyev Rovsen

#1 h Aug 15, 2014, 4:58 PM • 2 Y

Y by Adventure10, Mango247

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

professordad

4549 posts

professordad

#2 h Aug 15, 2014, 8:39 PM • 3 Y

Y by Pirkuliyev Rovsen, Adventure10, Mango247

Probably not the best way to do it

Let $O$ be the circumcenter. $AB = c$ , $AC = b$ , $BC = a$ . Let $BD = CD = x$ . Let $\angle BAC = A$ , $\angle ABC = B$ , $\angle ACB = C$ . Let $\angle AEB = \theta$ . Notice that $BOCD$ is cyclic, so $\angle CBD = \angle COD = A$ . Then $\cos{A} = \frac{a}{2x}$ . Now for the two-way proof:

$AB^2+AC^2 = 2BC^2 \Longrightarrow AE=DE$ :

From LOC we have $a^2=2bc\cos{A}$ . Then $\frac{a}{2\cos{A}} \cdot \sin{A} = \frac{bc\sin{A}}{a} = \frac{bc}{2R} = c\sin{B}$ . This becomes $x\sin{A} = c\sin{B}$ , so $\frac{c\sin{B}}{\sin{\theta}} = \frac{x\sin{A}}{\sin{\theta}}$ .

From LOS on $\triangle ABE$ and $\triangle BDE$ we have $\frac{\sin{B}}{AE} = \frac{\sin{\theta}}{c}$ and $\frac{\sin{A}}{DE} = \frac{\sin{\theta}}{x}$ , so $AE = \frac{c\sin{B}}{\sin{\theta}}$ and $DE = \frac{x\sin{A}}{\sin{\theta}}$ so $AE = DE$ .

$AE=DE \Longrightarrow AB^2+AC^2 = 2BC^2$ :

Let $\overline{AD} \cap O = A,X$ . Let $BE = m$ , $CE = n$ , with $m+n=a$ , and let $AE=DE=y$ , $EX = s$ .

By Stewart's on $\triangle ABD$ and $\triangle ACD$ we have
$\begin{align*}x^2+c^2&=2y^2+2m^2\\ x^2+b^2 &= 2y^2+2n^2\end{align*}$ Adding yields $2x^2+b^2+c^2=4y^2+2m^2+2n^2$ . By POP, $(y-s)(2y) = x^2$ and $sy = mn$ , so $2y^2-2mn=x^2$ . Combining this with the previous equation gives $b^2+c^2=2m^2+2n^2+4mn$ so $b+c=m+n=a$ .

Z K Y

N Quick Reply