Planegeometrien

Planegeometrien, widely referred to as plane geometry in English, is the branch of geometry that studies figures and properties on a two‑dimensional plane. It examines points, lines, angles, and shapes such as triangles, quadrilaterals, circles, and polygons, with notions of distance, angle measure, and area.

Core methods in planegeometrien include synthetic approaches, which derive results from axioms and logical reasoning, and analytic (coordinate) methods, which model figures with algebra. Ruler-and-compass constructions, such as constructing perpendicular bisectors or angle bisectors, are traditional tools. Fundamental objects include lines, angles, triangles, and circles, along with derived concepts like similarity and congruence, which underpin many theorems about shape and proportion.

Euclidean planegeometrien rests on a set of axioms, including the parallel postulate, which leads to familiar results such as the Pythagorean theorem, area formulas, and the congruence and similarity criteria for triangles. The parallel postulate also distinguishes Euclidean plane geometry from non‑Euclidean geometries, such as hyperbolic and elliptic geometries, where the nature of parallelism differs.

Key results in planegeometrien cover triangle properties (angle sum, side–angle relationships, and similarity/congruence criteria), area calculations for polygons, and relationships among circles (tangency, chords, and arc measures). Coordinate geometry extends the subject by representing geometric objects with equations, enabling algebraic methods for proving theorems and solving problems.

Historically, planegeometrien originated with ancient Greek geometry, was formalized in Euclid’s Elements, and later expanded by Descartes, Apollonius, and others. It remains foundational for fields such as architecture, engineering, computer graphics, and geographic information systems.