if for G centroid, AB+GC = AC+GB, then the triangle is isosceles

Art of Problem Solving

AoPS Online

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

if for G centroid, AB+GC = AC+GB, then the triangle is isosceles

G H J

Reveal topic tags

L

G H BBookmark kLocked kLocked NReply

Source: Spanish Mathematical Olympiad 1996 P2

The post below has been deleted. Click to close.

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

30629 posts

#1 h Jul 31, 2018, 10:08 AM • 1 Y

Y by Adventure10

Let $G$ be the centroid of a triangle $ABC$ . Prove that if $AB+GC = AC+GB$ , then the triangle is isosceles

Z K Y

The post below has been deleted. Click to close.

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

702 posts

#2 h Aug 1, 2018, 12:42 AM • 2 Y

Y by Adventure10, Mango247

Let $a,b,c$ be the length of sides $AB,BC,CA$ , respec-tively. By Stewart's theorem and the fact that G trisects each median(on the side further from the vertex), we deduce
$9GB^{2}=2a^{2}+2c^{2}-b^{2}, 9GC^{2}=2a^{2}+2b^{2}-c^{2}$
Assume $b>c$ . Assuming $AB+GC=AC+GB$ , we have
$3(b-c)=\sqrt{2a^{2}+2b^{2}-c^{2}}-\sqrt{2a^{2}+2c^{2}-b^{2}}=\frac{3(b-c)^{2}}{\sqrt{2a^{2}+2b^{2}-c^{2}}+\sqrt{2a^{2}+2c^{2}-b^{2}}}<\frac{3(b-c)^{2}}{\sqrt{2(b-c)^{2}+2b^{2}-c^{2}}+\sqrt{2(b-c)^{2}+2c^{2}-b^{2}}}$ since $a^2>(b-c)^2$ by the triangle inequality. However $2(b-c)^{2}+2b^{2}-c^{2}=(2b-c)^{2}$ , so we have $3(b-c)<\frac{3(b^{2}-c^{2})}{2b-c+\left | 2c-b \right |}$
If $b\leq2c$ then two sides are equal, a contracdition. If $b>2c$ we get $9(b-c)^{2}<3(b^{2}-c^{2})$ ; upon dividing off $3(b-c)$ and rerranging, we get $2b<4c$ , again a contracdition. Thus we cannot have $b>c$ or similarly $b<c$ , so $b=c$

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a