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This lecture introduces Rational Trigonometry, a modern alternative to traditional trigonometry that offers greater generality, power, and often simpler calculations. Explore the Cross Law, the fundamental formula of Rational Trigonometry, which relates quadrances and spreads (rational versions of distances and angles) in a triangle. Learn how this approach works within affine geometry over general fields with metrical structure imposed by symmetric bilinear forms. Discover how Rational Trigonometry extends beyond Euclidean geometry to Relativistic geometry and finite fields, with applications in projective geometry leading to novel mathematical directions. The content builds on the work initiated in the 2005 book "Divine Proportions: Rational Trigonometry to Universal Geometry" and requires an understanding of basic linear algebra.
