MathDB

Triangles, orthocenter and reflection, proving AF intersects middle point BC

Source: Azerbaijan 2022 JBMO TST G3.

July 5, 2022

geometrygeometric transformationreflection

Problem Statement

In acute, scalene Triangle

ABC

H

is orthocenter,

BD

and

CE

are heights.

X,Y

are reflection of

A

from

D

E

respectively such that the points

X,Y

are on segments

DC

and

EB

. The intersection of circles

HXY

and

ADE

F.

(

F \neq H

). Prove that

AF

intersects middle point of

BC

. (

M

in the diagram is Midpoint of

BC

)