First let's consider the figure on the left. We have from part 3 that triangles AHC', DB'E, and A'GF have equal areas. When we subtract the area of triangle A'B'C' from these three triangles, we see that the areas of quadrilaterals AB'C'E, IHB'A', and B'GFC' are equal.
From part 2 we know that the ratio of the areas of triangles A'B'C' and ABC is 1/4 (34 - 24 sqrt (2).
From part 3 we know that the ratio of the areas of triangles IHC' and ABC is 3 - 2 sqrt 2.
Thus, the ratio of the areas of quadrilateral IHB'A' and triangle ABC is 3 - 2 sqrt 2 - (/4 (34 - 24 sqrt (2)) = 4 sqrt (2) - 5.5.
Click here for a GSP sketch that verifies this relationship.
Now we will consider the figure on the left. In this figure, we want to find the area of one of the shaded regions. Let's find the area of region C'ECF. In the figure below, we can see that the the area of C'EFC is equal to the area of DGC minus the areas of DA'C'E, B'GFC', and A'B'C'.
By construction, we know that the ratio of the area of triangle A'B'C' to the area of triangle ABC is 1/2.
From above, we know that the ratio of the area of quadrilateral
DA'C'E to the area of triangle ABC is
4 sqrt (2) - 5.5.
The same is true for quadrilateral B'GFC'.
From part two, we know that the
ratio of the area of triangle A'B'C' to the area of triangle ABC is
1/4 (34 - 24 sqrt (2).
Thus, the ratio of the area of quadrilateral C'ECF to the area of triangle ABC is
1/2 - (4 sqrt (2) - 5.5) - (4 sqrt (2) - 5.5) - (1/4 (34 - 24 sqrt (2)).
This simplifies to
3 - 2 * sqrt (2).
The ratios for the other two shaded sections can be found in a similar manner, and they are the same.
Thus, the areas of the three shaded regions are the same.
Click here for a GSP sketch that verifies this relationship.