Finding the Centroid of a Triangle:

The centroid of a triangle is the point where the three medians of the triangle intersect. A median is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. To find the centroid, we need to find the midpoint of each side and then find the intersection point of the three medians.

Finding the Midpoint of a Side:

The midpoint of a line segment is the point that is halfway between the two endpoints. To find the midpoint of a line segment, we use the midpoint formula:

Midpoint = ((x1 + x2) / 2 , (y1 + y2) / 2)

where (x1, y1) and (x2, y2) are the coordinates of the endpoints.

Finding the Coordinates of the Centroid:

Once we have the midpoints of each side, we can find the equation of the three medians and then find their intersection point. The intersection point is the centroid of the triangle.

Let's find the midpoint of each side first:

Midpoint of AB = ((-1 + 5) / 2 , (-3 - 6) / 2) = (2, -4.5)
Midpoint of AC = ((-1 + 2) / 2 , (-3 + 3) / 2) = (0.5, 0)
Midpoint of BC = ((5 + 2) / 2 , (-6 + 3) / 2) = (3.5, -1.5)

Now we can find the equation of the medians. The equation of a line that passes through two points (x1, y1) and (x2, y2) is:

(y - y1) / (y2 - y1) = (x - x1) / (x2 - x1)

Let's find the equation of the median that passes through A:

Midpoint of BC = (3.5, -1.5)
Slope of BC = (-6 - 3) / (5 - 2) = -3
Equation of BC: y + 6 = -3(x - 5) => y = -3x + 21
Slope of median from A to BC = (3 - (-3)) / (2 - 5) = 2/3
Equation of median from A to BC:
y + 3 = (2/3)(x + 1) => y = (2/3)x + (5/3)

Similarly, we can find the equations of the medians that pass through B and C:

Equation of median from B to AC: y + 6 = (1/3)(x - 5) => y = (1/3)x + 11/3
Equation of median from C to AB: y - 3 = (-2/3)(x - 2) => y = (-2/3)x + 5/3

Now we need to find the intersection point of these three medians. We can solve the system of equations:

y = (2/3)x + (5/3)
y = (1/3)x + 11/3
y = (-2/3)x + 5/3

Solving these equations, we get x = -1 and y =