Geometry

Congruence and similarity criteria, and the special right-triangle ratios.

Medians, altitudes, angle bisectors, and midsegments — and where they meet.

Centroid, incenter, circumcenter, orthocenter, and how they line up.

Heron's formula, Brahmagupta's formula, and Stewart's theorem.

Chords, tangents, secants, and the Power of a Point relationships between them.

Inscribed angles, parallel-line angle pairs, and building angle-chase chains.

Supplementary opposite angles and Ptolemy's theorem for diagonals.

Equal tangent segments and the relationships that follow from them.

Lines, circles, and the shoelace formula for polygon area.

Reflections, rotations, translations, and homothety.

Volume and surface area for prisms, cylinders, cones, and spheres.

Ratio and concurrency tools for cevians and collinear points.

Interior and exterior angle formulas for convex and regular polygons.

Probability as a ratio of favorable to total length, area, or volume.

Advanced tools for tangent-circle chains and rotation-scaling configurations.