The tiger-gon

The final (of this set) Catriona Shearer puzzle.

20.
The Tiger-gon

What fraction is shaded? The hexagon is regular, with equally spaced dots around its perimeter.

Spoiler!

Divide the hexagon into triangles:

The top triangle shares a base with triangle , and both have their third vertex at the same perpendicular distance to that base, so the top triangle has the same area as triangle .

The middle triangle shares a base with triangle , and its third vertex has four times the perpendicular distance to that base, so the middle triangle has four times the area of triangle .

Similarly the bottom triangle has three times the area of one of the equilateral triangles.

The shaded area is then equal in area to eight equilateral triangles, and the hexagon consists of 24 such triangles, so the shaded area is of the hexagon.