Triangle A:
This Triangle Center is called the Centroid. It is the intersection points of the medians of the triangle sides. Using the coordinate graph as your guide, do you see any relationship between the distance from the centroid to the vertex and the centroid to the midpoint? Move the points around and see if you can determine a hypothesis.
Triangle B:
This Triangle Center is called the Incenter. It is the intersection point of the angle bisectors of the three triangle angles. Move the points of this triangle around so that you have an acute triangle, an obtuse triangle and a right triangle. What do you notice about the incenter with all three of these triangles. Take note of the relationship between the incenter and all three triangle sides.
Triangle C:
This Triangle Center is called the Orthocenter. It is the intersection point of the perpendicular bisectors of the three sides of the triangle. Move the points of this triangle around so that you have an acute triangle, an obtuse triangle, and a right triangle. What do you notice about the location of the orthocenter in these three types of triangles. Move the points around some more to make sure that your hypothesis holds.