Synthetic geometry
Synthetic geometry studies figures and their properties through axioms, definitions, and geometric constructions rather than coordinate equations.
Euclidean geometry: The geometry of points, lines, planes, and figures governed by Euclid’s postulates, including the parallel postulate. Synthetic geometry is most often practiced within Euclidean geometry’s axiomatic framework.
Geometric construction: The creation of geometric figures using specified tools and rules, classically an unmarked straightedge and compass. Constructed auxiliary points and lines expose relationships that a proof can use.
Analytic geometry: The study of geometry using coordinates and algebraic equations to represent points, figures, and transformations. It is the principal alternative to deriving results directly from geometric axioms and constructions.
Euclid's Elements: Euclid’s ancient Greek treatise presenting geometry and number theory through definitions, postulates, and propositions. Its axiomatic presentation became the enduring model for synthetic geometric proof.
Triangle geometry: The study of triangles and their centers, lines, circles, and metric or incidence relationships. Triangles provide the standard setting for many classical synthetic constructions and proofs.
Axiomatic system: A formal structure built from undefined terms, axioms, definitions, and rules of inference. Synthetic proofs derive geometric claims from explicitly stated assumptions.
Synthetic proof: A proof that derives geometric conclusions from axioms and figure relations without reducing them to coordinate equations. This is the characteristic reasoning process of synthetic geometry.
Cartesian coordinate system: A system that identifies points by ordered numerical coordinates relative to perpendicular axes. Synthetic geometry reasons about points relationally instead of assigning them numerical positions.
Euclid: An ancient Greek mathematician whose treatise Elements organized foundational geometry into a deductive system. His work supplied the classical framework from which synthetic geometry developed.
Circle geometry: The study of circles and their relations to points, lines, angles, and other circles. Synthetic arguments often exploit tangency, cyclicity, and equal angles in circle configurations.