Napoleon-Like Properties of Spherical Triangles

Mark R. Treuden

Department of Mathematics and Computing
University of Wisconsin-Stevens Point
Stevens Point, WI 54481-3897
USA
Phone: (715) 346-3734
m2treude@uwspmail.uwsp.edu

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ABSTRACT

If equilateral triangles are constructed outwards or inwards on the sides of any given triangle, 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 along 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) units perpendicular to line 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) units 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,B)/2 and t=(+-)d(A,B)/(2(sqrt(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 system (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.

Keyword(s): geometry, computer algebra systems, Mathematica