Geometry

Advanced and olympiad geometry: triangle centers and the classical theorems, power of a point, cyclic quadrilaterals, inversion, and non-Euclidean geometry.

Advanced Triangle Theorems

• Triangle Centers — Centroid, circumcenter, incenter, orthocenter — four triples of lines, four meeting points.
• Heron's Formula — Triangle area from three sides: A = √s(s-a)(s-b)(s-c).
• Ceva's Theorem — When three cevians of a triangle meet at a single point.
• Menelaus's Theorem — The collinearity test for a transversal cutting a triangle's sides.
• Simson Line — Perpendicular feet from a point are collinear exactly on the circumcircle.
• Nine-Point Circle — One circle through nine triangle points, centered on the Euler line.

Circles, Lattices & Inversion

• Power of a Point — Every line through P meets the circle in the same product: PX · PY is an invariant.
• Cyclic Quadrilaterals — Opposite angles supplementary, the concyclicity test, and the exterior angle.
• Ptolemy's Theorem — Diagonals and sides of a cyclic quadrilateral: pq = ac + bd.
• Pick's Theorem — Lattice-polygon area: A = i + b/2 - 1.
• Inversive Geometry (Inversion) — Inversion in a circle: the relation OP · OP' = r^2 and how it moves lines and circles.

Solids & Non-Euclidean

• Platonic Solids — Five solids, no more, no less — and Euler's polyhedral formula proves it.
• Non-Euclidean Geometry — Drop the parallel postulate and the angle sum changes.

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