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Kosovo Mathematical Olympiad, #4. (Grade 12)

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March 13, 2011
geometry proposedgeometry

Problem Statement

It is given a convex hexagon A1A2⋯A6A_1A_2 \cdots A_6 such that all its interior angles are same valued (congruent). Denote by a1=A1A2‾,  a2=A2A3‾, ⋯ ,a6=A6A1‾.a_1= \overline{A_1A_2},\ \ a_2=\overline{A_2A_3},\ \cdots , a_6=\overline{A_6A_1}.
a)a) Prove that holds: a1−a4=a2−a5=a3−a6 a_1-a_4=a_2-a_5=a_3-a_6 b)b) Prove that if a1,a2,a3,...,a6a_1,a_2,a_3,...,a_6 satisfy the above equation, we can construct a convex hexagon with its same-valued (congruent) interior angles.
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