Little Geometry(olympiad)

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

Little Geometry(olympiad)

G H J

Reveal topic tags

L

G H BBookmark kLocked kLocked NReply

Source: Olympiad in the World(2017)

The post below has been deleted. Click to close.

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

647 posts

#1 h Dec 7, 2017, 10:54 PM • 1 Y

Y by Adventure10

Let $ABC$ be a triangle with, $AB$ ≠ $AC$ , and let $K$ is your incenter. The points $P$ and $Q$ are the points of the intersections of the circumcicle( $BCK$ ) with the line(s) $AB$ and $AC$ , respectively. Let $D$ be intersection of $AK$ and $BC$ .
Show that $P, Q, D$ are collinears.

This post has been edited 3 times. Last edited by mathisreal, Jun 28, 2020, 1:33 AM

Z K Y

The post below has been deleted. Click to close.

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

62 posts

#2 h Dec 8, 2017, 5:20 AM • 2 Y

Y by Adventure10, Mango247

P,Q is a reflecton of C and B across line AK

Z K Y

The post below has been deleted. Click to close.

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

647 posts

#4 h Dec 10, 2017, 9:44 AM • 2 Y

Y by Adventure10, Mango247

Any solutions ?

Z K Y

The post below has been deleted. Click to close.

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

1594 posts

#5 h Dec 10, 2017, 9:57 AM • 2 Y

Y by Adventure10, Mango247

Replace $K$ with $I$ . It's well known that the center of $\odot(BIC)$ is $AI\cap\odot(ABC)$ . Therefore the reflection of $B$ across $AI$ lies on $\odot(BIC)$ . This reflection must lies on $AC$ too therefore it must be $Q$ . Similarly $P$ is the reflection of $C$ across $AI$ . Therefore lines $PQ$ and $BC$ are symmetric across $AI$ which implies the colinearity.

Z K Y

The post below has been deleted. Click to close.

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

30629 posts

#6 h Apr 21, 2020, 2:36 PM

Y by

Let $ABC$ be a triangle with $AB \ne AC$ , and let $K$ be the incenter. Let $P$ and $Q$ be the other points of the intersections of the circumcircle of triangle $BCK$ with the lines $AB$ and $AC$ , respectively. Let $D$ be intersection of $AK$ and $BC$ .
a) Show that $P, Q, D$ are collinear,
b) Let $T$ , diiferent from $P$ , be the intersection of the circumcircles of triangles $PQB$ and $QDC$ . Prove that $T$ lies on the circumcircle of the triangle $ABC$ .

[full wording with source: 2017 Brazil MO level 2 p5 - OBM - above was only the 1st part]

This post has been edited 3 times. Last edited by parmenides51, Apr 21, 2020, 2:46 PM

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a