Given triangle ABC, the medial triangle can be constructed by connecting the midpoints of the 3 sides of the triangle. To see a GSP sketch of the medial triangle, click here.
Notice that the area of the of the medial triangle is one-fourth the area of the orignal triangle. In addition, the two triangles are similar.
One can construct the centroid, the orthocenter, the circumcenter, and the incenter of these triangles. The centroid of a triangle is the intersection point of the three medians. The orthocenter is the intersection of the altitudes. The circumcenter is the center of the circumcircle. It can be constructed by finding the intersection of the perpendicular bisectors of the sides of the triangle. The incenter is the center of the inscribed circle. It can be constructed by finding the intersection point of the angle bisectors of the sides of the triangle.
To see a GSP sketch of these centers of triangle ABC and its medial triangle, click here.
What do you notice about these centers?
You should notice that the centroid is the same point for both triangle ABC and its medial triangle. In addition, the orthocenter of the medial triangle is the same point as the circumcenter of the original triangle. Also, the two incenters and the centroid appear to be colinear.