In a coordinate system, the midpoints of triangle segments show important relationships that help us understand triangles better.

1. Finding the Midpoint: The midpoint, which we can call $M$, of a line segment has two ends, $(x_1, y_1)$ and $(x_2, y_2)$. We find the midpoint using this formula:

   $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$

   This means we take the average of the x-coordinates and the average of the y-coordinates.

2. Understanding Slopes: The slope $m$ between two points helps us figure out if lines are parallel (running in the same direction) or perpendicular (meeting at right angles). For points $(x_1, y_1)$ and $(x_2, y_2)$, we use this formula:

   $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

   This tells us how steep a line is between two points.

3. Finding the Centroid: The centroid, or center point, of a triangle is called $G$. It is found by taking the average of the triangle's corners (or vertices) $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$. We find it like this:

   $$G = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$$

   This gives us the triangle's balance point.

Understanding these ideas helps us study triangles and solve geometry problems more easily.