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Geometry · Midpoints · Midline theorem · Parallelogram · Convex polygon

Problem 1, 1999

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RegionalOpen answer

A convex pentagon \(A_1A_2A_3A_4A_5\) is given. Let \(B_1\), \(B_2\), \(B_3\), \(B_4\) be the midpoints of the sides \(A_1A_2\), \(A_2A_3\), \(A_3A_4\), \(A_4A_5\), in that order, and let \(M\) be the midpoint of \(B_2B_4\) and \(N\) the midpoint of \(B_1B_3\). Determine the ratio of the lengths of the segments \(MN\) and \(A_1A_5\).

A1 A2 A3 A4 A5 B1 B2 B3 B4 M N
The four midpoints, the two segments \(B_1B_3\) and \(B_2B_4\), and their midpoints \(N\) and \(M\).

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Serbian Regional Competition (Okruzno takmicenje) 1999, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source

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