Electronic Proceedings of the Eighth Annual International Conference on Technology in Collegiate MathematicsHouston, Texas, November 16-19, 1995Paper C096Napoleon-Like Properties of Spherical Triangles |
Mark R. TreudenDepartment of Mathematics and Computing University of Wisconsin-Stevens Point Stevens Point, WI 54481-3897 USA Phone: (715) 346-3734 m2treude@uwspmail.uwsp.edu list of all papers by this author |
Click to access this paper: |
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