If a vertex of a triangle is `(1,1)` , and the middle points of two sides passing through it are `-2,3)` and `(5,2),` then find the centroid and the incenter of the triangle.

Explanation

To solve the problem step by step, we will find the coordinates of points B and C using the midpoint formula, then calculate the centroid and the incenter of the triangle.

### Step 1: Identify the given points
- Vertex A of the triangle: \( A(1, 1) \)
- Midpoint of side AB: \( M(-2, 3) \)
- Midpoint of side AC: \( N(5, 2) \)

### Step 2: Find the coordinates of point B
...