Triangle inside triangle which have common thinks

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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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 upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/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
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
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, 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
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
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
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
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

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
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
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
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
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
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

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

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
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)

0 replies

jlacosta

Yesterday at 11:16 PM

0 replies

Square problem

Jackson0423 1

N 3 minutes ago by maromex

Construct a square such that the distances from an interior point to the vertices (in clockwise order) are
1,2,3,4, respectively.

1 reply

Jackson0423

19 minutes ago

maromex

3 minutes ago

Sequence with infinite primes which we see again and again and again

Assassino9931 4

N 6 minutes ago by SimplisticFormulas

Source: Balkan MO Shortlist 2024 N6

Let $c$ be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence $a_1 = c$ , $a_{n+1} = a_n^3 + c$ .

4 replies

Assassino9931

Apr 27, 2025

SimplisticFormulas

6 minutes ago

Fermat points of Pentagon

Jackson0423 1

N 21 minutes ago by Jackson0423

It is known that, in general, a pentagon has three Fermat points. But I'm curious—if there are exactly two Fermat points inside the pentagon, under what conditions does the distance sum reach a minimum? Can you help me?

1 reply

Jackson0423

41 minutes ago

Jackson0423

21 minutes ago

Inequality , Exponent problem

biit 5

N 22 minutes ago by Jackson0423

If $\ P=(\frac {6375}{6374})^ {6374}$ , $\ Q=(\frac {6375}{6374})^ {6375}$ then prove that $P^{Q}$ > $Q^{P}$

5 replies

biit

41 minutes ago

Jackson0423

22 minutes ago

Maximum value

Ecrin_eren 2

N an hour ago by Ecrin_eren

a,b,c are positive real numbers such that
(a+b)^2 (a+c)^2=16abc
What is the maximum value of a+b+c

2 replies

Ecrin_eren

3 hours ago

Ecrin_eren

an hour ago

How many pairs

Ecrin_eren 0

an hour ago

Let n be a natural number and p be a prime number. How many different pairs (n, p) satisfy the equation:

p + 2^p + 3 = n^2 ?

0 replies

Ecrin_eren

an hour ago

0 replies

Find the domain and range of $f(x)=2-|x-5|.$

Vulch 2

N 2 hours ago by jasperE3

Find the domain and range of $f(x)=2-|x-5|.$

2 replies

Vulch

Yesterday at 2:07 AM

jasperE3

2 hours ago

Maximum value of n

Ecrin_eren 1

N 2 hours ago by jasperE3

"Let M be the set {1, 2, 3, ..., 2025}. Jack selects n different subsets of M. If the union of any two subsets Jack selects is never equal to M, what is the maximum possible value of n?"

1 reply

Ecrin_eren

4 hours ago

jasperE3

2 hours ago

Inequalities

sqing 7

N 2 hours ago by sqing

Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$(1-abc) (1-a)(1-b)(1-c) \ge 208$ $(1+abc) (1-a)(1-b)(1-c) \le -224$ $(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840$

7 replies

sqing

Jul 12, 2024

sqing

2 hours ago

How many integer pairs

Ecrin_eren 0

4 hours ago

"Let m and n be integers. How many different integer pairs (m, n) satisfy the equation m^3 - 3m^2n + 4n^3 = 44?"

0 replies

Ecrin_eren

4 hours ago

0 replies

How many triangles

Ecrin_eren 0

5 hours ago

"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?"

0 replies

Ecrin_eren

5 hours ago

0 replies

All possible values of k

Ecrin_eren 1

N 6 hours ago by Ecrin_eren

The roots of the polynomial
x³ - 2x² - 11x + k
are r₁, r₂, and r₃.

Given that
r₁ + 2r₂ + 3r₃ = 0,
what is the product of all possible values of k?

1 reply

Ecrin_eren

Today at 8:42 AM

Ecrin_eren

6 hours ago

Angle AEB

Ecrin_eren 1

N Today at 10:25 AM by Ecrin_eren

In triangle ABC, the lengths |AB|, |BC|, and |CA| are proportional to 4, 5, and 6, respectively. Points D and E lie on segment [BC] such that the angles ∠BAD, ∠DAE, and ∠EAC are all equal. What is the measure of angle ∠AEB in degrees?

1 reply

Ecrin_eren

Today at 9:26 AM

Ecrin_eren

Today at 10:25 AM

20 fair coins are flipped, N of them land heads 2024 TMC AIME Mock #6

parmenides51 6

N Today at 9:26 AM by MelonGirl

$20$ fair coins are flipped. If $N$ of them land heads, find the expected value of $N^2$ .

6 replies

parmenides51

Apr 26, 2025

MelonGirl

Today at 9:26 AM

Triangle inside triangle which have common thinks

Ege_Saribass 1

N Apr 27, 2025 by Ege_Saribass

Source: Own

An acute triangle $\triangle{ABC}$ is given on the plane. Let the points $D$ , $E$ and $F$ be on the sides $BC$ , $CA$ and $AB$ , respectively. ( $D$ , $E$ and $F$ are different from the vertices $A$ , $B$ and $C$ ) Also the points $X$ , $Y$ and $Z$ are taken such that $DZEXFY$ is an equilateral hexagon which the opposite sides are parallel. Suppose that the circumcenters of $\triangle{ABC}$ and $\triangle XYZ$ are coincident. Then determine the least possible value of:
$\frac{A(\triangle{XYZ})}{A(\triangle{ABC})}$ Note: $A(\triangle{KLM}) =$ area of $\triangle{KLM}$

1 reply

Ege_Saribass

Apr 26, 2025

Ege_Saribass

Apr 27, 2025

Triangle inside triangle which have common thinks

G H J

Reveal topic tags

G H BBookmark kLocked kLocked NReply

Source: Own

The post below has been deleted. Click to close.

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

Ege_Saribass

29 posts

Ege_Saribass

#1 h Apr 26, 2025, 5:24 PM

Y by

This post has been edited 1 time. Last edited by Ege_Saribass, Apr 27, 2025, 11:10 AM
Reason: missing information

Z K Y

The post below has been deleted. Click to close.

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

Ege_Saribass

29 posts

Ege_Saribass

#2 h Apr 27, 2025, 12:40 PM

Y by

Answer

$\frac{1}{4}$

Solution

Let $O$ be the circumcenter of $\triangle XYZ$ and $\triangle ABC$ .
Claim 1: $O$ is the orthocenter of $\triangle DEF$ .
Proof: By parallel sides, we can easily conclude that $DE // XY$ . Also by isosceles $\triangle XFY$ , we have $OF \perp XY$ . Hence $OF \perp DE$ . Similarly, $OE \perp DF$ and $OD \perp EF$ . Thus, we get the claim. $\square$

We can also conclude that $\triangle DEF \sim \triangle XYZ$ . Moreover, $\triangle EXY \cong \triangle YDZ$ hence $EF = YZ$ thus $\triangle XYZ \cong \triangle DEF$ . Hence, we are going to find the least possible value of $\frac{A(\triangle{DEF})}{A(\triangle{ABC})}$ .

Claim 2: $\triangle DEF \sim \triangle ABC$

Proof: Now we know that the circumcenter of $\triangle ABC$ is also the orthocenter of $\triangle DEF$ by Claim 1.
Let's use this. Take inversion around $(ABC)$ . Let any point $K$ goes to $K'$ via inversion.

Now we get $(AOBF')$ , $(AOCE')$ , $(COBD')$ .
Also $F'$ and $O$ are at the opposite sides according to the line $AB$ . Similar things satisfy for $D'$ and $E'$ .
By basic angle chasing, one can conclude $\angle AF'B + \angle BD'C + \angle CE'A = 180$
Also it is well known that by the inversion, $O$ is the incenter of $\triangle D'E'F'$ . Because $O$ was the orthocenter of $\triangle DEF$ .

Subclaim: $F'AE'$ are linear.
Suppose not. If $[E'F']$ intersects $AO$ , $A$ is outside the $\triangle D'E'F'$ .
Also we know that $\angle AF'O = \angle BF'O$ and $\angle E'F'O = \angle D'F'O$ . Thus, $B$ is outside the $\triangle D'E'F'$ too. Similarly, $C$ is outside the $\triangle D'E'F'$ too.
So we get $\angle AF'B > \angle E'F'D'$ , $\angle BD'C > \angle F'D'E'$ and $\angle CE'A > \angle D'E'F'$ . The sum of this three inequality gives us $180 > 180$ which is a contradiction.
If $[E'F']$ doesn't intersect $AO$ , we get $A$ , $B$ and $C$ is inside the $\triangle D'E'F'$ and same contradiction similarly.
And the proof of the subclaim is done. $\square$

Because of inversion and subclaim, we got $(FAEO)$ . Similarly, $(DOFB)$ and $(DOEC)$ . And by simple angle chasing, we get $\angle A = \angle EDF$ . Similarly $\angle B = \angle DEF$ and $\angle C = \angle EFD$ . Hence, the proof of the Claim 2 is done. $\square$

Now let the circumradius of the $\triangle DEF$ be $r$ and circumradius of the $\triangle ABC$ be $R$ .
By the Claim 2, $\frac{A(\triangle{DEF})}{A(\triangle{ABC})} = \frac{r^2}{R^2}$

Claim 3: $R \leq 2r$
Proof: By law of sines at $(DEF)$ and $(AFOE)$ , these circles have same radius. Let $P$ be the circumcenter of $(AFOE)$ .
By triangle inequality, $AO \leq AP + PO \implies R \leq 2r$
Hence, we have the claim. $\square$

By the Claim 3, we have
$\frac{r^2}{R^2} \geq \frac{1}{4}$ hence $\frac{A(\triangle{DEF})}{A(\triangle{ABC})} \geq \frac{1}{4}$ .

If we choose $D$ , $E$ and $F$ the midpoints in $\triangle ABC$ , it is obvious that we get the equality case and everything holds.
$\blacksquare$

This post has been edited 4 times. Last edited by Ege_Saribass, Apr 30, 2025, 1:00 PM
Reason: e4 e5 Nf3 Nc6

Z K Y

N Quick Reply