A triangle ABC has a trisected angle A. The angle trisectors divide side a (opposite angle A) into three segments which are BD, DE, and DC. The lengths are 2, 3, and 6, respectively. What are the lengths of the other sides?

I think you should apply the angle bisector theorem here, since AD bisects angle ABE and AE bisects angle ADC.

To find the lengths of the other two sides of triangle ABC, we can use the Law of Sines. This law states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

First, let's label the angles in triangle ABC. Angle A is trisected, so let's call the three resulting angles α, β, and γ. Angle B will be opposite to side b, angle C will be opposite to side c, and angle A will be opposite to side a. So, we have:

Angle A = α
Angle B = β
Angle C = γ

Now, let's label the sides of the triangle:
Side a = DE = 3
Side b = DC = 6
Side c = BD = 2

According to the Law of Sines, we have the following ratios:

a/sin(α) = b/sin(β) = c/sin(γ)

We know the length of side a and the angles α and β, so we can find the ratio b/sin(β). Rearranging the formula, we have:

b/sin(β) = a/sin(α)

Plugging in the values we know:

b/sin(β) = 3/sin(α)

To find side b, we need to find the value of sin(β) and sin(α).

From the triangle, we can see that sin(β) = DE/BC and sin(α) = DE/BD.

Plugging in the values we know:

sin(β) = (DE / BC) = (3 / (BC + BD)) = 3 / (6 + 2) = 3/8
sin(α) = (DE / BD) = (3 / 2)

Now we can substitute these values back into the equation:

b/sin(β) = 3/sin(α)

b/(3/8) = 3 / (3/2)

b/(3/8) = 2

Multiplying both sides by 3/8:

b = 2 * (3/8) = 6/8 = 3/4

So, the length of side b is 3/4.

Now that we know the length of side b, we can use the Law of Sines to find side c.

c/sin(γ) = b/sin(β)

c/sin(γ) = (3/4) / (3/8) = (3/4) * (8/3) = 8/4 = 2

Now, we know that c = 2.

In summary, the lengths of the sides of triangle ABC are:
Side a = DE = 3
Side b = DC = 3/4
Side c = BD = 2