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Divided Sides

Source: ILL 1970 - Problem 13.

May 24, 2011
geometry unsolvedgeometry

Problem Statement

Each side of an arbitrary △ABC\triangle ABC is divided into equal parts, and lines parallel to AB,BC,CAAB,BC,CA are drawn through each of these points, thus cutting △ABC\triangle ABC into small triangles. Points are assigned a number in the following manner: (1)(1) A,B,CA,B,C are assigned 1,2,31,2,3 respectively (2)(2) Points on ABAB are assigned 11 or 22 (3)(3) Points on BCBC are assigned 22 or 33 (4)(4) Points on CACA are assigned 33 or 11 Prove that there must exist a small triangle whose vertices are marked by 1,2,31,2,3.
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