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A,X,Y,E are concyclic iff AB=BC, given BC/AC=3/2

Source: 2023 Croatia MO p7 HMO https://artofproblemsolving.com/community/c1251476_croatia_mo_hmo___geometry

May 5, 2024

geometryConcyclicisosceles

Problem Statement

Given is an acute-angled triangle

ABC

in which holds

|BC|: |AC| = 3:

2. Let

D

be the midpoint of the side

\overline{AC}

, and P the midpoint of the segment

\overline{BD}

. A point

X

is given on the line

AC

so that

|AX| = |BC|

, where

A

is between

X

and

C

. The line

XP

intersects the side

\overline{BC}

at point

E

. The line

DE

intersects the line

AP

at point

Y

. Prove that the points

A

,

X

,

Y

,

E

lie on one circle if and only if

|AB| = |BC|

.

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