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equal ratios of sidelengths on similar triangles lead to collinear points

Source: Sharygin 2006 finals 9.3

September 12, 2019

equal ratiosimilar trianglescollineargeometry

Problem Statement

Triangles

ABC

and

A_1B_1C_1

are similar and differently oriented. On the segment

AA_1

, a point

A'

is taken such that

AA' / A_1A'= BC / B_1C_1

. We similarly construct

B'

and

C'

. Prove that

A', B',C'

lie on one straight line.