Dreiecks

Dreiecks, commonly known in English as triangles, are plane figures with three sides and three vertices. In Euclidean geometry, the sum of interior angles is exactly 180 degrees. The three sides must satisfy the triangle inequality: the length of any side is less than the sum of the other two.

Triangles are classified by side lengths: equilateral (all sides equal), isosceles (two equal sides), scalene (all different). By angles: acute (all angles less than 90 degrees), right (one angle exactly 90 degrees), obtuse (one angle greater than 90 degrees).

Area and perimeter: the perimeter is the sum of the side lengths. Area can be computed as base times height divided by two, or using Heron's formula from the side lengths, or as one half of the product of two sides times the sine of the included angle.

Congruence and similarity: triangles are congruent if their corresponding sides and angles match (criteria include SSS, SAS, ASA, AAS, and, for right triangles, HL). Similarity uses equal corresponding angles (AA) and proportional corresponding sides (SSS, SAS).

Key points and lines: medians intersect at the centroid; altitudes meet at the orthocenter; angle bisectors meet at the incenter; perpendicular bisectors meet at the circumcenter. These centers have standard properties and relations within the triangle.

Applications: triangles underpin much of geometry, trigonometry, and surveying. They appear in architecture and computer graphics and serve as a fundamental building block for polygons and polyhedra.