a triangle construction, angle, opposite side and median of other side

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jlacosta

2 hours ago

0 replies

k a My Retirement & New Leadership at AoPS

rrusczyk 1571

N Mar 26, 2025 by SmartGroot

I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE

1571 replies

rrusczyk

Mar 24, 2025

SmartGroot

Mar 26, 2025

Number theory

Maaaaaaath 0

a few seconds ago

Let $m$ be a positive integer . Prove that there exists infinitely many pairs of positive integers $(x,y)$ such that $\gcd(x,y)=1$ and :

$xy | x^2+y^2+m$

0 replies

Maaaaaaath

a few seconds ago

0 replies

Only consecutive terms are coprime

socrates 35

N 18 minutes ago by zuat.e

Source: 7th RMM 2015, Problem 1

Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$ ?

35 replies

socrates

Feb 28, 2015

zuat.e

18 minutes ago

D1019 : Dominoes 2*1

Dattier 4

N 21 minutes ago by Dattier

I have a 9*9 grid like this one:

IMAGE

We choose 5 white squares on the lower triangle, 5 black squares on the upper triangle and one on the diagonal, which we remove from the grid.
Like for example here:

IMAGE

Can we completely cover the grid remove from these 11 squares with 2*1 dominoes like this one:

IMAGE

4 replies

Dattier

Mar 26, 2025

Dattier

21 minutes ago

inequalities hard

Cobedangiu 4

N 23 minutes ago by Cobedangiu

problem

.................

4 replies

Cobedangiu

Mar 31, 2025

Cobedangiu

23 minutes ago

No more topics!

a triangle construction, angle, opposite side and median of other side

parmenides51 2

N Dec 24, 2023 by parmenides51

Source: 2003 Oral Moscow Geometry Olympiad grade 9 p1

Construct a triangle given an angle, the side opposite the angle and the median to the other side (researching the number of solutions is not required).

2 replies

parmenides51

Oct 26, 2020

parmenides51

Dec 24, 2023

a triangle construction, angle, opposite side and median of other side

G H J

Reveal topic tags

geometry

construction

G H BBookmark kLocked kLocked NReply

Source: 2003 Oral Moscow Geometry Olympiad grade 9 p1

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parmenides51

30629 posts

parmenides51

#1 h Oct 26, 2020, 8:07 PM • 1 Y

Y by GeoKing

Construct a triangle given an angle, the side opposite the angle and the median to the other side (researching the number of solutions is not required).

This post has been edited 1 time. Last edited by parmenides51, Oct 28, 2020, 6:38 AM
Reason: removed original wording

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RagvaloD

4894 posts

RagvaloD

#2 h Oct 27, 2020, 6:45 AM • 1 Y

Y by GeoKing

Let $AB$ is this side, $\alpha$ - this angle, $m_c$ - median and $M$ is midpoint of $AB$
Easy to build triangle $ABC$ with $\angle CAB=90, \angle CBA=90-\alpha$ Now build circle $\omega_1$ - circumcircle of $ABC$ and circle $\omega_2$ - circle with center $M$ and radius $m_a$ . Let $D$ is point of intersection $\omega_1,\omega_2$ Then $ABD$ is triangle, which we want to build.

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parmenides51

30629 posts

parmenides51

#3 h Dec 24, 2023, 2:27 PM • 1 Y

Y by GeoKing

another construction using centroid

Click to reveal hidden text

Let BL be the Known median, M be the midpoint of BC=a and G be the centroid of ABC

- Construct segment BC=a
- Construct an angle.<BAC=<A
- Locus of A are two arcs of circles with center M and radii R such that they "see" segment BC under angle <A (one at each side of AB) without the endpoints B,C

-Because MG is homothetic MA with center M and ratio MG/MA=1/3, locus of G are two arcs.of circles. with center M and radius R/3, (homothetic to the previous arcs, such that one at each side of AB) without the endpoints B,C

- We draw the circle (B, BL/3) i.e. with center B and radius BL/3 wich intersects one of the arcs of the locus of G at a point, let it be G'

- Triangle BCG' has been constructed

- Finally we extended MG' beyond G'to segment G'A' =2MG'

- Triangle A' BC is the wanted one

Latex when I use my pc

This post has been edited 2 times. Last edited by parmenides51, Dec 24, 2023, 2:40 PM

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