page_title: Rational trigonometry
section_title: Triple quad formula
hierarchical_section_title: Rational trigonometry / Laws of rational trigonometry / Triple quad formula
caption_attribution_description: English: Diagram used to illustrate the proof of the rational trigonometry Triple Quad Formula. Generated from an AutoCAD drawing drawn by Derrick Oswald.
context_page_description: Rational trigonometry is a proposed reformulation of metrical planar and solid geometries by Canadian mathematician Norman J. Wildberger, currently a professor of mathematics at the University of New South Wales. His ideas are set out in his 2005 book Divine Proportions: Rational Trigonometry to Universal Geometry. According to New Scientist, part of his motivation for an alternative to traditional trigonometry was to avoid some problems that he claims occur when infinite series are used in mathematics. Rational trigonometry avoids direct use of transcendental functions like sine and cosine by substituting their squared equivalents. Wildberger draws inspiration from mathematicians predating Georg Cantor's infinite set-theory, like Gauss and Euclid, who he claims were far more wary of using infinite sets than modern mathematicians. To date, rational trigonometry is largely unmentioned in mainstream mathematical literature.
context_section_description: The three points A₁, A₂, A₃ are collinear if and only if:
  where Q₁, Q₂, Q₃ represent the quadrances between A₁, A₂, A₃ respectively. It can either be proved by analytic geometry (the preferred means within rational trigonometry) or derived from Heron's formula, using the condition for collinearity that the triangle formed by the three points has zero area.
  Proof (click at right to show/hide)
  The line AB has the general form:
  where the (non-unique) parameters a, b, c can be expressed in terms of the coordinates of points A and B as:
  so that, everywhere on the line:
  But the line can also be specified by two simultaneous equations in a parameter t, where t = 0 at point A and t = 1 at point B:
  or, in terms of the original parameters:
  If the point C is collinear with points A and B, there exists some value of t (for distinct points, not equal to 0 or 1), call it λ, for which these two equations are simultaneously satisfied at the coordinates of the point C, such that:
  Now, the quadrances of the three line segments are given by the squared differences of their coordinates, which can be expressed in terms of λ:
  where use was made of the fact that (−λ + 1)² = (λ − 1)².
  Substituting these quadrances into the equation to be proved:
  Now, if A and B represent distinct points, such that a² + b² ≠ 0, we may divide both sides by Q(AB)² = (a² + b²)²:
caption: Illustration of nomenclature used in the proof.
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