In triangle ABC, D bisects side BC, G bisects side AB, and the points E and F trisect side AC. What is the area of the shaded polygon, if the area of ABC is 108?

I tryed to find the area of triangles on the side and I figured that if you put a line from the top of the triangle to Point F in the triangle it should be parallel to one of the sides.

To find the area of the shaded polygon, we need to determine the areas of individual triangles within the triangle ABC.

Let's start by labeling the points as shown in the figure below:

A
/\
/ \
G /____\ E
/ \ / \
/___\/___\
B D C


We know that points D and G bisect sides BC and AB, respectively. This means that line segment DG divides the triangle ABC into two smaller triangles, namely triangle ADG and triangle DGB. Similarly, line segment GE divides triangle ADG into two more triangles, namely triangle AEG and triangle DEG.

Given that the area of triangle ABC is 108, we need to determine the areas of triangles ADG, DGB, AEG, and DEG.

To find the area of triangle ADG, we can use the fact that line segment DG is a median, which means it divides triangle ABC into two equal areas. So, the area of triangle ADG is half the area of triangle ABC, which is 108/2 = 54.

Next, we can find the area of triangle DGB. Since line segment DG is a median, it also divides triangle DGB into two equal areas. Thus, the area of triangle DGB is also 54.

Moving on to triangle AEG, we know that point F trisects side AC. This means that line segment EF divides triangle AEG into two equal areas. Since line segment EF passes through point G, we can conclude that triangle GEG is congruent to triangle GEA. Therefore, the area of triangle AEG is half the area of triangle GEA.

Similarly, triangle DEG can be divided into two equal areas by line segment EF. Hence, the area of triangle DEG is also half the area of triangle GEA.

Since triangle GEA is formed by trisecting side AC, we can say that the area of this triangle is one-third of the area of triangle ABC. Thus, the area of triangle GEA is 108/3 = 36.

Therefore, the areas of triangles AEG and DEG are 36/2 = 18 each.

Now, if we combine the shaded polygon with triangles AEG and DEG, we can see that it is a quadrilateral formed by the sides of triangle ABC and line segment EF connecting points E and F.

To find the area of this quadrilateral, we subtract the areas of triangles AEG and DEG from the area of triangle ABC. So, the area of the shaded polygon is 108 - (18 + 18) = 72.

Hence, the area of the shaded polygon is 72.

To find the area of the shaded polygon, let's first divide triangle ABC into smaller triangles.

Let's label the points as follows:
- A, B, C: Vertices of triangle ABC
- D: Point of the bisector of BC
- G: Point of the bisector of AB
- E, F: Points trisecting side AC

Now, let's divide triangle ABC into smaller triangles:
- Triangle AED: This triangle is equivalent to one-third of the area of triangle ABC since E divides side AC into three equal segments.
- Triangle EFD: This triangle is equivalent to one-third of the area of triangle ABC since F divides side AC into three equal segments.
- Triangle GEB: This triangle is equivalent to half of the area of triangle ABC since G bisects side AB.

So, the shaded polygon is formed by the areas of triangles AED, EFD, and GEB.

To find the area of the shaded polygon, subtract the sum of the areas of triangles AED, EFD, and GEB from the area of triangle ABC:

Area of shaded polygon = Area of ABC - (Area of AED + Area of EFD + Area of GEB)

Given that the area of triangle ABC is 108, substitute the known values:

Area of shaded polygon = 108 - (1/3 * 108 + 1/3 * 108 + 1/2 * 108)

Simplifying the expression:

Area of shaded polygon = 108 - (36 + 36 + 54)

Area of shaded polygon = 108 - 126

Area of shaded polygon = -18

Therefore, the area of the shaded polygon is -18. Please note that a negative area does not make geometric sense, so please double-check the given information and calculations made.