triangle

A triangle is a polygon with three sides and three interior angles. It is the simplest geometric polygon and forms the basic unit of more complex shapes. The sum of its interior angles in Euclidean geometry is 180 degrees.

Classification by sides: equilateral (all sides equal), isosceles (two sides equal), scalene (no equal sides). Classification by angles: acute (all angles < 90), obtuse (one angle > 90), right (one angle = 90).

Key properties: the triangle inequality states that the sum of any two side lengths exceeds the third. The area can be computed as base times height divided by two; Heron's formula also exists: area = sqrt(s(s-a)(s-b)(s-c)) where s is semiperimeter. For right triangles, Pythagoras' theorem holds: a^2 + b^2 = c^2.

Centers and lines: medians intersect at the centroid, which balances the triangle; perpendicular bisectors intersect at the circumcenter; angle bisectors intersect at the incenter; altitudes intersect at the orthocenter. Each of these centers has a corresponding circle: circumcircle and incircle.

Congruence and similarity: triangles are congruent if corresponding sides and angles match; common criteria include SSS, SAS, ASA, AAS, and HL for right triangles. Similar triangles have proportional sides and equal corresponding angles; criteria include AA, SSS, SAS.

Applications: trigonometry arises from right triangles; laws of sines and cosines relate side lengths to angles in any triangle. Triangles are fundamental in geometry, surveying, architecture, and computer graphics.