MathDB

AX bisects < CAB , 2 midpoints, foot of altitude, 2 circumcircles related

Source: Mathematics Regional Olympiad of Mexico Northeast 2017 P2

September 21, 2022

geometryangle bisector

Problem Statement

Let

ABC

be a triangle and let

N

and

M

be the midpoints of

AB

and

CA

, respectively. Let

H

be the foot of altitude from

A

. The circumcircle of

ABH

intersects

MN

P

, with

P

and

M

on the same side relative to

N

, and the circumcircle of

ACH

intersects

MN

Q

, with

Q

and

N

on the same side relative to

M

BP

and

CQ

intersect at

X

. Prove that

AX

is the angle bisector of

\angle CAB