Medians of round triangles

We'll begin by generalizing the concept of median to round triangles. In the metric geometries of Euclidean, hyperbolic, and elliptic geometries, a median can be defined as a straight line that connects a vertex of the triangle to the midpoint of the opposite side. Inversive geometry is not a metric geometry, so midpoints cannot be defined. By use of the dual vertices of a triangle, however, we can define medians in such a way that they specify to the usual concept of median in these three subgeometries. Furtheremore, the three medians of a triangle will all meet at one point, which we'll call the centroid of the triangle.

In order to find the median from vertex A to the opposite side BC, we'll construct two circles which will act as lines parallel to the sides AB and AC, and connect A to their intersection. More precisely, draw the circles A*B*C and A*BC* as drawn in cyan in the diagram. These two circles intersect at two points, A* and another. Then the circle through these two points and A gives the median of the triangle from A through the opposite side BC. Similarly, we can construct the medians through B and C. These three medians are drawn in green in the diagram. All three pass through two points G and G*, and we'll call these two points the centroids of the triangle ABC and its dual A*B*C*.

Medians and Centroids

When the triangle is Euclidean, that is, when A*, B*, and C* coincide as a pole P, then one of the two centroids is the pole P, and the other is the expected centroid of the triangle. The hyperbolic case is displayed below. Note that the cyan circles aren't hyperbolic lines, but the green ones are.

Centroids of a Hyperbolic Triangle

Note that in this hyperbolic case, if the vertices are not all inside the bounding disk, or all outside the bounding disk, then the centroids are purely imaginary.

And here is the centroid for an elliptic triangle, again treated as a special case of an inversive triangle. Note that the cyan circles aren't ellliptic lines, but the green ones are.

Centroids of an Elliptic Triangle

David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610

Email: djoyce@clarku.edu
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