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Hexagon with an equal area distribution

Source: Pan African MO 2013 Q3

June 30, 2013

geometrygeometric transformationreflectiontrigonometrygeometry unsolved

Problem Statement

Let

ABCDEF

be a convex hexagon with

\angle A= \angle D

and

\angle B=\angle E

. Let

K

and

L

be the midpoints of the sides

AB

and

DE

respectively. Prove that the sum of the areas of triangles

FAK

KCB

and

CFL

is equal to half of the area of the hexagon if and only if

\frac{BC}{CD}=\frac{EF}{FA}.