line AO passes through the midpoint of segment EF

Art of Problem Solving

AoPS Online

High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

3 M G

BBookmark VNew Topic kLocked

High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

line AO passes through the midpoint of segment EF

G H J

Reveal topic tags

L

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.

26 posts

#1 h Mar 30, 2025, 5:08 AM

Y by

Given a triangle $ABC$ with $AB < AC$ and the angle bisector $AD$ .
The line passing through $A$ and perpendicular to $AC$ intersects the line passing through $B$ and parallel to $AD$ at point $E$ .
The line passing through $A$ and perpendicular to $AB$ intersects the line passing through $C$ and parallel to $AD$ at point $F$ .
Let $O$ be the intersection of the three perpendicular bisectors of triangle $ABC$ .
Prove that line $AO$ passes through the midpoint of segment $EF$ .

Z K Y

The post below has been deleted. Click to close.

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

469 posts

#2 h Mar 30, 2025, 3:04 PM

Y by

We have $\triangle ABE\sim\triangle ACF$ because $\angle ABE=\frac12\angle BAC=\angle ACF$ and $\angle BAE=90^{\circ}-\angle BAC=\angle CAF$ . Afterwards, using the complex plane with origin $O$ , let $e-a=k(b-a)$ and $f-a=\overline k(c-a)$ . We have $kb+\overline kc=0$ since $\angle BAC+2\angle BAE=180^{\circ}$ , yielding $\frac{e+f}2=a-ak-a\overline k$ . This is a multiple $a$ , so $\frac{e+f}2$ lies on line $AO$ .

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a