Geoff C Smith, Statics and the Moduli Space of Triangles,
Forum Geometricorum 5 (2005) 181--190.
Abstract: The variance of a weighted collection of points is used to prove
classical theorems of geometry concerning homogeneous quadratic functions
of length (Apollonius, Feuerbach, Ptolemy, Stewart) and to deduce some of
the theory of major triangle centers. We also show how a formula for the
distance of the incenter to the reflection of the centroid in the nine-point
center enables one to simplify Euler's method for the reconstruction of a
triangle from its major centers. We also exhibit a connection between Poncelet's
porism and the location of the incenter in the circle on diameter GH (the
orthocentroidal or critical circle). The interior of this circle is the moduli
(classification) space of triangles.
[ps file][pdf file]