THE DIFFERENT CENTERS OF TRIANGLES

Venki Ramachandran & Roberta Kirkham

1. The Centroid G is the intersection of the three medians of a triangle. The median being a line joining the midpoint of a side with the opposite vertex.

2. The Orthocenter of a triangle is the point of intersection of the altitudes dropped from each vertex to the opposite side

The Orthocenter could lie inside or outside the triangle!

3. The Circumcenter of a triangle is the point of intersection of the perpendicular bisectors of the sides. A circle drawn with the circumcenter as the center would pass through the three vertices of the triangle

The circumcenter C of a trianlge may lie within or outside of the triangle. It can however be observed that the locus of C will be one of the three perpendicular bisectors.

4. The INCENTER of a triangle is the point inside the triangle that is equidistant from the three sides. Such a point would obviously lie on the angle bisector of the vertices.

Needless to say, irrespective of the shape of the triangle, the incenter lies inside the circle. Further, a circle drawn with the Incenter as as center and radius equal to the perpendicular distance to any one side, will touch the other two sides.

5. To construct G,H, C and I for a given triangle and observe them

The Circumcenter C, the Centroid G and the Orthocenter are always collinear. When the triangle becomes an equilateral triangle, all these four points become concurrent. Further, the ratio of the distances between C, G and H is constant.

6. The Medial triangle and the relationship between it's C,G,H and I to the parent triangle.

The medial triangle is constructed and the following are noticed. G, the centroid of the parent triangle remains the G, centroid of the Medial triangle. Next, a strange thing is observed; the Circumcenter of the parent triangle is the Orthocenter of the Medial triangle.