MathDB

Vietnam MO 2016 P6

Source: VMO 2016

July 29, 2016

geometryperpendicular bisectorcircumcircleincenterangle bisector

Problem Statement

Given a triangle

ABC

inscribed by circumcircle

(O)

. The angles at

B,C

are acute angle. Let

M

on the arc

BC

that doesn't contain

A

such that

AM

is not perpendicular to

BC

.

AM

meets the perpendicular bisector of

BC

at

T

. The circumcircle

(AOT)

meets

(O)

at

N

(

N\ne A

).
a) Prove that

\angle{BAM}=\angle{CAN}

.
b) Let

I

be the incenter and

G

be the foor of the angle bisector of

\angle{BAC}

.

AI,MI,NI

intersect

(O)

at

D,E,F

respectively. Let

{P}=DF\cap AM, {Q}=DE\cap AN

. The circle passes through

P

and touches

AD

at

I

meets

DF

at

H

(

H\ne D

).The circle passes through

Q

and touches

AD

at

I

meets

DE

at

K

(

K\ne D

). Prove that the circumcircle

(GHK)

touches

BC

.

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