Chromogeometry and Relativistic Conics

If Q1 = Q(A2,A3), Q2 = Q(A1,A3) and Q3 = Q(A1,A2) are the quadrances of a triangle A1A2A3, then Pythagoras’ theorem and its converse can together be stated as: A1A3 is perpendicular to A2A3 precisely when Q1 + Q2 = Q3. Figure 1 shows an example where Q1 = 5, Q2 = 20 and Q3 = 25. As indicated for the large square, these areas may also be calculated by subdivision and (translational) rearrangement, followed by counting cells. There is a sister theorem—the Triple quad formula—that Euclid did not know, but which is fundamental for rational trigonometry, introduced in [2]: A1A3 is parallel to A2A3 precisely when