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This paper shows how a recent reformulation of the basics of classical geom-etry and trigonometry reveals a three-fold symmetry between Euclidean and non-Euclidean (relativistic) planar geometries. We apply this chromogeometry to look at conics in a new light. Pythagoras, area and quadrance To measure a line segment in the plane, the ancient Greeks measured the area of a square constructed on it. Algebraically, the parallelogram formed by a vector v = A1A2 = (a, b) and its perpendicular B (v) = (−b, a) has area Q = det µ a b −b a = a2 + b2. (1) The Greeks referred to building squares as ‘quadrature’, and so we say that Q is the quadrance of the vector v, or the quadrance Q (A1, A2) between A1 and A2. This notion makes sense over any field. A