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3 lines parallel wanted, symmetric of vertices wrt opposite sides

Source: 2007 Oral Moscow Geometry Olympiad grades 8-9 p5

October 18, 2020

geometrySymmetricparallelmidpoint

Problem Statement

Given triangle

ABC

. Points

A_1,B_1

and

C_1

are symmetric to its vertices with respect to opposite sides.

C_2

is the intersection point of lines

AB_1

and

BA_1

. Points

A_2

and

B_2

are defined similarly. Prove that the lines

A_1 A_2, B_1 B_2

and

C_1 C_2

are parallel.
(A. Zaslavsky)