Read a diagram, justify every relation and build a complete argument. Begin at the first unfamiliar idea; the prerequisite links help you find a shorter route. You can open any lesson.

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  1. G01

    Angle chasing

    Find an angle using stated incidence and justified equalities.

    Inside: Lines and angle relationships · Parallel lines · Triangle angle sum · Angle-chasing with reasons

  2. G02

    Triangle congruence

    Prove a useful triangle correspondence before transferring equalities.

    Inside: Correspondence and SSS/SAS · ASA/AAS and RHS/HL · What SSA and AAA cannot prove · Auxiliary triangles

  3. G03

    Triangle inequalities

    Use strict triangle inequalities and explain geometric equality limits.

    Inside: Side-length feasibility · Side-angle comparison · Broken paths and extremal distances

  4. G04

    Triangle area ratios

    Translate a length ratio into an area ratio using an identified common altitude.

    Inside: Same-altitude and same-base triangles · Area ratios and shared regions

  5. G05

    The midpoint theorem

    Prove a midpoint or a parallel segment from the appropriate direction of the theorem.

    Inside: Midpoint theorem and converse · Midpoint constructions

  6. G06

    Parallel lines and angle bisectors

    Use parallelism or an angle bisector to obtain a correctly oriented ratio.

    Inside: Parallel intercepts and converse · Internal angle bisectors · External angle bisectors

  7. G07

    Similar triangles

    Identify a correspondence and distinguish length scaling from area scaling.

    Inside: AA, SAS and SSS similarity · Scale factors · Area ratios · Hidden similar triangles

  8. G08

    Pythagoras, medians and Stewart’s theorem

    Choose a length identity from a verified geometric configuration.

    Inside: Pythagoras and its converse · Acute and obtuse tests · Apollonius and median lengths · Stewart’s theorem

  9. G09

    Quadrilaterals and their diagonals

    Use the defining properties of each quadrilateral without assuming extra ones.

    Inside: Parallelogram tests · Rectangles, rhombi and squares · Trapezia and kites · Diagonal and midpoint arguments

  10. G10

    Concurrency and collinearity

    Distinguish concurrent lines from collinear points and select a suitable criterion.

    Inside: Centres and elementary concurrency · Carnot’s criterion · Ceva and its converse · Menelaus and directed ratios · Pappus as an advanced extension

  11. G11

    Circles, tangents and power of a point

    Combine angle and length properties while checking the point’s position.

    Inside: Chords, arcs and tangent facts · Alternate segment theorem · Power of a point · Intersecting chords and tangent-secant products · Radical axes and centres · Common tangents

  12. G12

    Cyclic and tangential quadrilaterals

    Prove cyclicity or tangency before applying its consequences.

    Inside: Cyclic criteria and converses · Ptolemy and its extension · Simson–Wallace line · Tangential quadrilaterals

  13. G13

    Trigonometry in geometry

    Select a trigonometric relation that simplifies the geometry and check angle domains.

    Inside: Sine rule and circumradius · Cosine and projection rules · Napier and Mollweide identities · Half-angle formulae · Triangle area and Heron connections · Triangle centres, inradius and exradii · Quadrilateral areas · Regular polygons

  14. G14

    Constructing triangles

    Give construction steps and prove they produce exactly the required possibilities.

    Inside: Basic ruler-and-compass constructions · Loci from side and angle data · Existence, uniqueness and multiple solutions

These original lessons introduce the methods and give practice with solutions. A single short session is not a full assessment of Olympiad readiness. Build depth through the written challenges and your country’s official past papers.