Centers of a Triangle


Play with the files above for a while. It looks like the medians of the triangle in the Centroid file meet in a single point, the altitudes of the triangle in the Orthocenter file meet in a single point, and the perpendicular bisectors of the triangle in the Circumcenter file meet in a single point. Can we prove this?
First, we need a way to describe a general triangle. Given any triangle, place coordinate axes on it so that one edge rests on the xaxis, and one vertex lies at the origin. Label the other two points (2a,0) and (2b,2c), and label the edges e_{1}, e_{2}, and e_{3}, as shown:
By wisely choosing where we put the origin and onto which edge we put the xaxis, we can assume (without loss of generality) that a, b, and c are all nonnegative. In addition, note that if c = 0, we have a degenerate triangle in the form of a line along the xaxis, and if a = 0, we have a degenerate triangle in the form of a line segment along the yaxis. So we can assume that, if we are in the case of a nondegenerate triangle, we have a > 0, b ≥ 0, and c > 0. Now that we have a way of describing the triangle, let's show that the centroid, orthocenter, and circumcenter are welldefined points. That is, let's prove that the three lines defining each actually meet in a single point (a different point for each). Click here to continue.
