Problem:
Let ABCDEF be a regular hexagon. Let G,H,I,J,K, and L be the midpoints of sides AB,BC,CD,DE,EF, and AF, respectively. The segments AH,BI,CJ,DK,EL, and FG bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of ABCDEF be expressed as a fraction nm where m and n are relatively prime positive integers. Find m+n.
Solution:
Without loss of generality, let AB=2, and place ABCDEF in the first and second quadrants of the coordinate plane with A=(0,0) and B=(2,0). Then C=(3,3),E=(0,23),F=(−1,3),G=(1,0), H=(25,23), and L=(−21,23). Then line AH has equation y=53x, line FG has equation y=2−3x+23, and line EL has equation y=(33)x+23. The intersection of lines AH and FG is then X=(75,73), and the intersection of lines EL and FG is Y=(7−3,753). Then XY is a side of the smaller hexagon, and the ratio of the areas is the square of the ratio of the sides, which is