Triangle Centers

The following tasks show how to find various triangle centers using Geometer's Sketchpad

Construct the triangle, construct the midpoints on the segments, construct segments connecting the midpoints to the vertices.

Construct the triangle, construct the angle bisectors, find the point of intersection, construct a perpendicular with the point of intersection and a side, construct a segment using the intersection point of the angle bisectors and the perpendicular with the side (this is the radius), then finally construct the circle.

Construct a triangle, construct the midpoints, construct perpendicular lines using the midpoints, find the point of intersection, construct a circle using that point and one of the vertices of the triangle.

What three special triangle centers lie on the same line no matter how you change the length of the sides of the triangle?