Geometric proof

A deductive argument that establishes a geometric claim from definitions, constructions, axioms, and previously proved results.

Axiom: A foundational statement accepted as a starting point within a formal system. Geometric proofs begin from accepted assumptions such as Euclid’s postulates.
Deductive reasoning: Reasoning that derives conclusions from premises by valid rules of inference. Each conclusion in a geometric proof must follow from stated premises or earlier results.
Pythagorean theorem: A theorem stating that a right triangle’s hypotenuse squared equals the sum of its legs squared. Its many proofs illustrate distinct ways to establish one geometric claim.
Non-Euclidean geometry: Geometries whose axioms differ from Euclid’s, especially in the treatment of parallel lines. Changing a geometric postulate can change which claims are provable.
Definition: A precise statement specifying the meaning of a term. Definitions fix what geometric objects and relations mean before an argument uses them.
Congruence (geometry): A relation between figures that have the same shape and size. Congruent triangles transfer equal sides and angles to establish further claims.
Euclid’s Elements: An ancient Greek mathematical treatise that organizes geometry through definitions, postulates, and propositions. Its propositions provide a foundational collection of classical geometric arguments.
Neutral geometry: Geometry developed from assumptions common to Euclidean and hyperbolic geometry, without a parallel postulate. It separates results that need no parallel-line assumption from those that do.
Theorem: A statement established by proof from axioms, definitions, and earlier results. A geometric proof turns a proposed claim into a theorem.
Similarity (geometry): A relation between figures with equal corresponding angles and proportional corresponding lengths. Similarity proofs establish proportionality without requiring figures to have equal size.

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