MathDB

Projective geometry

Source: 2017 VMO Problem 7

January 11, 2017

geometrycircumcircle

Problem Statement

Given an acute triangle

ABC

and

(O)

be its circumcircle. Let

G

be the point on arc

BC

that doesn't contain

O

of the circumcircle

(I)

of triangle

OBC

. The circumcircle of

ABG

intersects

AC

at

E

and circumcircle of

ACG

intersects

AB

at

F

(

E\ne A, F\ne A

).
a) Let

K

be the intersection of

BE

and

CF

. Prove that

AK,BC,OG

are concurrent.
b) Let

D

be a point on arc

BOC

(arc

BC

containing

O

) of

(I)

.

GB

meets

CD

at

M

,

GC

meets

BD

at

N

. Assume that

MN

intersects

(O)

at

P

nad

Q

. Prove that when

G

moves on the arc

BC

that doesn't contain

O

of

(I)

, the circumcircle

(GPQ)

always passes through two fixed points.

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