Introduction:
To find the coordinates of points of trisection on a line segment joining two given points, we can use the concept of section formula. The section formula states that if two points (x1, y1) and (x2, y2) divide a line segment in the ratio m:n, then the coordinates of the point dividing the line segment in the given ratio can be found using the formula:

x = (mx2 + nx1) / (m + n)
y = (my2 + ny1) / (m + n)

Given information:
The given points are (4, -1) and (-2, -3).

Step 1: Determine the ratio of trisection:
To find the coordinates of points of trisection, we need to determine the ratio in which the line segment is divided. In this case, we want to find the trisection points, which means the line segment is divided into three equal parts. Therefore, the ratio of trisection is 1:1:1.

Step 2: Apply the section formula:
Using the section formula, we can substitute the given values into the formula to find the coordinates of the points of trisection.

For the x-coordinate:
x = (mx2 + nx1) / (m + n)
= (1*(-2) + 1*4) / (1 + 1)
= (4 - 2) / 2
= 2 / 2
= 1

For the y-coordinate:
y = (my2 + ny1) / (m + n)
= (1*(-3) + 1*(-1)) / (1 + 1)
= (-3 - 1) / 2
= -4 / 2
= -2

Step 3: Find the coordinates of the points of trisection:
The coordinates of the points of trisection are (1, -2). As the line segment is divided into three equal parts, there will be two points of trisection, one at the 1/3rd distance and the other at 2/3rd distance.

Conclusion:
The coordinates of the points of trisection on the line segment joining (4, -1) and (-2, -3) are (1, -2). These points divide the line segment into three equal parts.