We start with a triangle PQR. 

First, we draw the median of the triangle through R 
1. Construct the bisector of the line segment PQ. Label the midpoint of the line S.
See Constructing a perpendicular bisector of a line segment 

2. Draw the median from the midpoint S to the opposite vertex R 

Next, we draw the second median of the triangle through P 
3. In the same manner, construct T, the midpoint of the line segment QR. See Constructing a perpendicular bisector of a line segment 

4. Draw the median from the midpoint T to the opposite vertex P 

(Optional step) Repeat for the third side. This will convince you that the three medians do in fact intersect at a single point.
But two are enough to find that point. 
5. Done. The point C where the two medians intersect is the centroid of the triangle PQR. 
