Areas of Hexagons

Draw a hexagon. In each case below, find the ratio of the area of this hexagon to the new one formed by the crossing lines inside it.

  1. Connect each corner to the corner two locations clockwise.

  2. Connect the midpoint of each side to the midpoints of the adjacent sides.
  3. Connect the midpoint of each side to the midpoints of the non-adjacent sides (but not straight across.)

Source: Original.

Solutions were received from Joseph DeVincentis, Jongmin Baek, Colin Bown, Bob Odineal, Radu Ionescu, Henry Bottomley, Sudipta Das, Denis Borris. Colin's answer is a nice simple summary:

I like the pure construction method:

  1. After drawing the diagram one can add a point in the center and draw connecting lines to the verticies of the smaller hexagon. Now look! 18 triangles of equal area. 6 are in the smaller hexagon. Ratio = 3:1.
  2. If the obtuse triangles from (1) are refected about their long edge one gets the included diagram. 24 triagles of equal area. 18 in the smaller haxagon. Ratio = 4:3.
  3. The hexagon in question is embedded in the diagram. 24 triagles of equal area. 6 in the smaller haxagon. Ratio = 4:1.
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