Starting with an equilateral triangle, the corners are cut off to form a regular hexagon. What is the ratio of the hexagon to the area of the original triangle?
The easiest way would be to draw an equilateral triangle 3 units on each side.
Call the original triangle ABC.
Divide each side into thirds.
Label the two points of trisection of AB as U and V.
Label the two points of trisection of BC as W and X.
Label the two points of trisection of CA as Y and Z.
Draw VW, XY and ZU to get hexagon UVWXYZ.
Draw UX, VY, and WZ.
They will meet at O, the center of the triangle.
You now have 9 congruent trianges, 6 in the hexagon.
The ratio of the hexagon to the area of the original triangle: 6 : 9 = 2 : 3.
You can also find the areas of the triangles and get the same answer that way.