Aarav’s Geometry

A rigorous course in Euclidean plane and solid geometry

Aarav’s Geometry

A rigorous course in Euclidean plane and solid geometry, built from postulates and proven step by step.

This course treats geometry the way Euclid did: as a system of careful reasoning starting from a small number of self-evident assumptions. Every result you’ll meet has been proven, by you or for you, from these foundations. The goal is not to memorize formulas — it is to learn to see why the world of shapes must be the way it is.

Primary sources: Kiselev’s Geometry (translated by Alexander Givental) and Richard Rusczyk’s Introduction to Geometry from the Art of Problem Solving.

The Course

Unit 1 — Logic & ProofsWhat it means to prove something. The five Euclidean postulates. Two-column proofs. Indirect proofs.

Unit 2 — Lines, Angles, ParallelsAngle pair relationships, the parallel postulate at work, and theorems on transversals and parallel lines.

Unit 3 — Triangles & CongruenceTriangle basics, congruence criteria, and the proof patterns that let one triangle transfer knowledge to another.

Unit 4 — Triangle Inequalities & CentersWhy triangle sides and angles constrain each other, plus medians, altitudes, bisectors, and classical centers.

Unit 5 — ConstructionsStraightedge-and-compass moves: copying segments and angles, bisecting, perpendiculars, parallels, and loci.

Unit 6 — Quadrilaterals & PolygonsParallelograms, rectangles, rhombi, trapezoids, regular polygons, and angle sums from proof rather than memory.

Unit 7 — SimilarityScale without distortion: ratios, similar triangles, proportionality, and the geometric meaning of enlargement.

Unit 8 — Right Triangles & TrigonometryPythagorean reasoning, special right triangles, altitude relations, and trigonometric ratios as geometry.

Unit 9 — CirclesChords, tangents, arcs, inscribed angles, cyclic quadrilaterals, and the power relationships hidden in circles.

Unit 10 — Area & PerimeterArea as a theorem-driven measure: polygons, circles, composite figures, and formulas with reasons behind them.

Unit 11 — Solid Geometry: PolyhedraPlanes, prisms, pyramids, angle relationships in space, and the structure of polyhedra.

Unit 12 — Solid Geometry: Volume & SurfaceVolume and surface area for solids, including cylinders, cones, spheres, and Cavalieri-style reasoning.

Unit 13 — TransformationsReflections, rotations, translations, symmetry, and congruence seen as motion of the plane.

Unit 14 — Coordinate GeometryBringing algebra to the plane: distance, midpoint, slope, equations of lines, and coordinate proofs.

This course is for Aarav, with a copy of Kiselev’s text beside him.