Euclidean geometry from classical mechanics

Abstract

Euclidean geometry is axiomatic in classical mechanics. In our paper, we have reversed the traditional roles of geometry and mechanics to obtain elegant results in Euclidean geometry. The convex or barycentric coordinates (α1, α2, α3) of point ? in the closed triangular region ?V1 V2V3 may be interpreted as weights α1, α2, α3 which, when placed at the vertices V1 V2 V3 respectively, determine ? as the balance point. Taking Archimedes' Law of Levers as our axiom, we obtain the convex coordinates for various special points of ?V1 V2 V3. Among these special points are the centroid, incenter, Gergonne point, and Nagel point. We then describe the construction of a uniform, calibrated plyboard triangle which serves as a physical model for ?V1 V2 V3. At each vertex, V1 we can suspend a weight Wi. By the proper selection of weights ?W1 W2 W3 we can move the balance point ? to any desired point of the interior of the triangle. The model serves as an excellent teaching device which emphasizes in a striking manner t...