Curriculum · Geometry
Geometry
From angle chasing to power of a point — training the eye to see the auxiliary line, the hidden similar triangle, the circle nobody drew.
Angles & Triangles
Angle Chasing
Parallel lines, triangle sums, isosceles symmetry — the deductive game every geometer plays first.
- Chase angles through parallel lines
- Use triangle and polygon angle sums
- Exploit isosceles triangles
The Triangle Toolkit
Congruence, the triangle inequality, medians, altitudes, and area formulas.
- Prove triangles congruent
- Apply the triangle inequality
- Compute area five ways
prerequisite: angle-chasing
Special Right Triangles & Pythagoras
45-45-90, 30-60-90, Pythagorean triples, and distance in disguise.
- Recall special ratios instantly
- Recognize Pythagorean triples
- Drop the right altitude
prerequisite: triangle-toolkit
Scaling & Circles
Similar Triangles
AA similarity and ratio bookkeeping — the single most-used tool in contest geometry.
- Spot AA similarity in configurations
- Track ratios through similar pairs
- Use similarity to compute lengths
prerequisite: triangle-toolkit
Circles & Inscribed Angles
Central and inscribed angles, arcs, tangents, and cyclic quadrilaterals.
- Apply the inscribed angle theorem
- Use tangent-radius perpendicularity
- Recognize cyclic quadrilaterals
prerequisite: angle-chasing
Power of a Point
Chord-chord, secant-secant, tangent-secant — one theorem, three costumes.
- State power of a point in all forms
- Choose the right configuration
- Combine with similarity
prerequisite: circle-theorems, similar-triangles
Analytic & Solid
Coordinate Methods
When synthetic fails, coordinatize: distance, area by Shoelace, clever placement.
- Place figures to simplify algebra
- Compute areas with the Shoelace formula
- Know when to coordinatize
prerequisite: coordinate-plane-basics, special-triangles
Area Techniques
Decomposition, ratios of areas, and same-base-same-height reasoning.
- Decompose complex regions
- Relate areas through shared heights
- Use ratio arguments
prerequisite: triangle-toolkit
3D Geometry
Volumes, surface areas, cross-sections, and unfolding tricks.
- Compute standard volumes
- Reason about cross-sections
- Unfold surfaces for shortest paths
prerequisite: special-triangles
Transformations
Reflections, rotations, translations — symmetry as a solving tool.
- Reflect to straighten paths
- Rotate to align structures
- Compose transformations
prerequisite: coordinate-geometry-methods
Trigonometry for Contests
Law of sines and cosines, area by sine, and identity fluency.
- Apply laws of sines and cosines
- Compute area via ½ab·sin C
- Use key identities
prerequisite: special-triangles