The Geodesic Center of Doubly-Covered Triangles

Allison Baird

Using Cinderella, a geometric modeling program, we explored finding the geodesic center of a polyhedron -- the point closest to all other surface points – in the simplest of all possible cases: a doubly-sided triangle.

Any three non-collinear points in the plane define a unique circle. The vertices of a triangle therefore create a unique circle; its center is known as the circumcenter. In an acute triangle, where no angle has measure greater than 90 degrees, the circumcenter lies within the triangle’s borders. The vertices of the triangle are equidistant from the circumcenter, and all other points of the triangle have a distance from the circumcenter that is less than the circle’s radius. Therefore, the circumcenter is the geodesic center of the doubly-sided acute triangle (Fig. 1a). Furthermore, in an acute isosceles triangle, the circumcenter lies on the perpendicular bisector of the side joining the angles of equal measure (Fig. 1b).

1a1b

Fig. 1a: the geodesic center (heavy black point) of an acute doubly-sided triangle, located at the center of the circle (in red) defined by the triangle's vertices.
Fig. 1b: the geodesic center of an acute isosceles doubly-sided triangle.

In an obtuse triangle, the circumcenter lies outside the triangle. By definition, however, the geodesic center must lie on or within the triangle’s borders. The center is the bisector of the triangle’s longest side (Figs. 2a, 2b). It is a conjecture that the perpendicular bisecetor of a chord on a circle passes through the center of that circle. Therefore, the geodesic center also happens to be the point on the triangle nearest to the circumcenter.

2a2b

Figs. 2a, 2b: the geodesic center of an obtuse doubly-sided triangle lies at the midpoint of the longest side; the circumcenter (small red point) is on the line perpendicular to this side at this point.