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We study the geometry of a Euclidean triangle from a Mobius geometric point of view. It turns out that its in- and ex-centers can be constructed in a symmetric and Mobius invariant way. We relate this construction to Thurston's center of symmetry of an ideal tetrahedron in hyperbolic space and discuss some implications for the Euclidean triangle.

The set of osculating circles of a given curve in S 3 forms a lightlike curve in the set of oriented circles in S 3 . We show that its “ 1 -dimensional measure” with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.