Ruler and Compass Construction 6
Hexagon on a Given Segment

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1. Circle centered at A, through B
2. Circle centered at B, through A
3. Intersection O, center of hexagon
4. Circle centered at O, through A
5. Intersections, points C and F
6. Circle centered at C, through O
7. Circle centered at D, through O
8. Hexagon sides BC, CD, DE, EF, FA

Why it works:
ABO forms an equilateral triangle, so chord AB subtends a 60 degree angle from O. Hence copying
the chord around the circle centered at O gives 60 degree chords, which form the hexagon.