MathDB

Arc Midpoints Form Cyclic Quadrilateral

Source: ISL 2022/G2

July 9, 2023

geometryarc midpointcyclic quadrilateralISL 2022

Problem Statement

In the acute-angled triangle

ABC

, the point

F

is the foot of the altitude from

A

, and

P

is a point on the segment

AF

. The lines through

P

parallel to

Y \ne A

and

AB

meet

ABD

at

D

and

E

, respectively. Points

X \ne A

and

Y \ne A

lie on the circles

ABD

and

ACE

, respectively, such that

DA = DX

and

EA = EY

. Prove that

B, C, X,

and

Y

are concyclic.

Back to Problems View on AoPS