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PQRS is a parallelogram if and only if ACBD is cyclic

Source: Austrian Mathematical Olympiad 1997, Part 2, D1, P3

July 4, 2011

geometryparallelogramgeometry proposed

Problem Statement

Let be given a triangle

ABC

. Points

P

on side

AC

and

Y

on the production of

CB

beyond

B

are chosen so that

Y

subtends equal angles with

AP

and

PC

. Similarly,

Q

on side

BC

and

X

on the production of

AC

beyond

C

are such that

X

subtends equal angles with

BQ

and

QC

. Lines

YP

and

XB

meet at

R

,

XQ

and

YA

meet at

S

, and

XB

and

YA

meet at

D

. Prove that

PQRS

is a parallelogram if and only if

ACBD

is a cyclic quadrilateral.

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