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Hexagon within the hexagon

Source: 2010 AIMEII Problem 9

April 1, 2010
ratiogeometryanalytic geometrytrigonometryrotationsymmetrynumber theory

Problem Statement

Let ABCDEF ABCDEF be a regular hexagon. Let G G, H H, I I, J J, K K, and L L be the midpoints of sides AB AB, BC BC, CD CD, DE DE, EF EF, and AF AF, respectively. The segments AH AH, BI BI, CJ CJ, DK DK, EL EL, and FG FG bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of ABCDEF ABCDEF be expressed as a fraction mn \frac {m}{n} where m m and n n are relatively prime positive integers. Find m \plus{} n.
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