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.

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