MathDB

JBMO Shortlist 2022 G2

Source: JBMO Shortlist 2022

June 26, 2023

geometryJuniorBalkanshortlistparallelogramsconcentric

Problem Statement

Let

ABC

be a triangle with circumcircle

k

. The points

A_1, B_1,

and

C_1

on

k

are the midpoints of arcs

\widehat{BC}

(not containing

A

),

\widehat{AC}

(not containing

B

), and

\widehat{AB}

(not containing

C

), respectively. The pairwise distinct points

A_2, B_2,

and

C_2

are chosen such that the quadrilaterals

AB_1A_2C_1, BA_1B_2C_1,

and

CA_1C_2B_1

are parallelograms. Prove that

k

and the circumcircle of triangle

A_2B_2C_2

have a common center. Comment. Point

A_2

can also be defined as the reflection of

A

with respect to the midpoint of

B_1C_1

, and analogous definitions can be used for

B_2

and

C_2

.

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