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two traingles and 6 points of intersection

Source: Iran Third Round MO 1997, Exam 2, P2

March 23, 2004

geometryparallelogramgeometry proposed

Problem Statement

Let

ABC

and

XYZ

be two triangles. Define

A_1=BC\cap ZX, A_2=BC\cap XY,

B_1=CA\cap XY, B_2=CA\cap YZ,

C_1=AB\cap YZ, C_2=AB\cap ZX.

Hereby, the abbreviation

g\cap h

means the point of intersection of two lines

g

and

h

.
Prove that

\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA}

holds if and only if

\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}

.

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