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Napoleon-Like Properties of Spherical Triangles

M. Treuden
Journal / Anthology

Proceedings of the 8th Annual ICTCM
Year: 1995

If equilateral triangles are constructed outwards or inwards on the sides of any given triangle, then the centroids of these triangles are the vertices of an equilateral triangle. In elementary Euclidean geometry this result is known as Napoleon's Theorem. Consider the following generalization of this construction. Let d(.,.) denote Euclidean distance and suppose A,B,C are the vertices of any given, positively oriented triangle. Let point X be located s units from A line AB and t units perpendicular to line AB. Assume s,t are directed distances with S measured positively from A to B and t positive when measured outward from triangle ABC. With the same sign conventions, the point Y is located s.d(B,C)/d(A,B) units from point B along line BC and t.d(B,C)/d(A,B) unites perpendicular to BC. Similarly, point Z is located s.d(C,A)/d(A,B) units from point C along line CA and t.d(C,A)/d(A,B) unites perpendicular to line CA. In this way the points X,Y,Z are proportionately positioned relative to the points A,B,C. Note that X,Y,Z are the centroids mentioned in Napoleon's Theorem when s=d(A,B0/2 and t=+d(A,B0/2/3). Are there other real numbers s,t for which triangle XYZ is equilateral? The answer to this question can be discovered by most any college geometry student aided with a computer algebra systems (CAS). In this paper we adapt the foregoing construction to certain classes of spherical triangles and use a CAS to determine various values of s,t with the properties given above.