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Not so hard geometry problem

Source: 2017 VMO Problem 3

January 11, 2017

geometrycircumcircle

Problem Statement

Given an acute, non isoceles triangle

ABC

and

(O)

be its circumcircle,

H

its orthocenter and

E, F

are the feet of the altitudes from

B

and

C

, respectively.

AH

intersects

(O)

at

D

(

D\ne A

).
a) Let

I

be the midpoint of

AH

,

EI

meets

BD

at

M

and

FI

meets

CD

at

N

. Prove that

MN\perp OH

.
b) The lines

DE

,

DF

intersect

(O)

at

P,Q

respectively (

P\ne D,Q\ne D

).

(AEF)

meets

(O)

and

AO

at

R,S

respectively (

R\ne A, S\ne A

). Prove that

BP,CQ,RS

are concurrent.

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