The problem asks to find the values for I, JK, L, M, N, O, PQ, R, S, T, U, V, and W, based on the given information about triangles and their areas. We are given that the sum of the areas of triangles ABD and ACD is equal to the area of triangle ABC. This leads to finding AD. Then, using the angle $\angle DAE$, we need to find the length of AE. Finally, we are asked to find the inradius $r_1$ of triangle ADC and the inradius $r_2$ of triangle ADE. Unfortunately, the image doesn't provide enough information about the triangle ABC to calculate the values. The expressions for the lengths and the radii cannot be derived from the given information only.

1. Problem Description

The problem asks to find the values for I, JK, L, M, N, O, PQ, R, S, T, U, V, and W, based on the given information about triangles and their areas. We are given that the sum of the areas of triangles ABD and ACD is equal to the area of triangle ABC. This leads to finding AD. Then, using the angle $\angle DAE$, we need to find the length of AE. Finally, we are asked to find the inradius $r_1$ of triangle ADC and the inradius $r_2$ of triangle ADE. Unfortunately, the image doesn't provide enough information about the triangle ABC to calculate the values. The expressions for the lengths and the radii cannot be derived from the given information only.

2. Solution Steps

Since we don't have information about sides and angles, it is impossible to solve this problem.

3. Final Answer

It is impossible to provide a final answer as there is not enough information given about triangle ABC in the prompt.