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.