Trekanter

Trekanter is the Danish term for triangles, a fundamental class of polygons in geometry. A triangle has three sides, three vertices, and three interior angles. The sum of its interior angles is always 180 degrees. Triangles are the simplest polygons and provide a basis for understanding more complex shapes; any polygon can be divided into triangles in a process called triangulation.

Triangles are commonly classified by their side lengths: equilateral (all sides equal), isosceles (two sides equal), and scalene (no equal sides). They are also classified by their angles: acute (all angles less than 90 degrees), right (one angle exactly 90 degrees), and obtuse (one angle greater than 90 degrees).

Key properties include the triangle inequality, which states that the sum of the lengths of any two sides exceeds the third. Area can be computed by base times height divided by two, by Heron's formula using all three side lengths, or by 1/2 times a side times the corresponding height. In a coordinate setting, the area can be calculated from the vertex coordinates with a determinant-based formula. Right triangles satisfy the Pythagorean theorem: a^2 + b^2 = c^2.

Triangles also exhibit congruence and similarity. Two triangles are congruent if they have matching side lengths and angles (criteria include SSS, SAS, ASA, AAS, and HL for right triangles). Similarity is established by AA, SSS, or SAS criteria, indicating proportional corresponding sides and equal corresponding angles.

Trigonometry in triangles uses the Law of Sines and Law of Cosines to relate sides and angles. Triangles contain notable centers and circles, including the centroid, incenter, circumcenter, and orthocenter, and they form the basis for many constructions and applications in surveying, engineering, computer graphics, and architecture.

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