MathDB

Vietnamese geo with way too many points

Source: VMO 2023 day 1 P4

February 24, 2023

geometry

Problem Statement

Given is a triangle

ABC

and let

D

be the midpoint the major arc

BAC

of its circumcircle. Let

M , N

be the midpoints of

AB , AC

and

J , E , F

are the touchpoints of the incircle

(I)

of

\triangle ABC

with

BC, CA, AB

. The line

MN

intersects

JE , JF

at

K , H

respectively;

IJ

intersects the circle

(BIC)

at

G

and

DG

intersects

(BIC)

at

T

.
a) Prove that

JA

passes through the midpoint of

HK

and is perpendicular to

IT

.
b) Let

R, S

respectively be the perpendicular projection of

D

on

AB, AC

. Take the points

P, Q

on

IF , IE

respectively such that

KP

and

HQ

are both perpendicular to

MN

. Prove that the three lines

MP , NQ

and

RS

are concurrent .

Back to Problems View on AoPS