Three angles always sum to a straight line—on a flat plane
A triangle is the simplest polygon: three sides, three vertices, three interior angles. In Euclidean geometry those angles always add to 180 degrees, or π radians—a fact equivalent to the parallel postulate. Area is half base times height, and a menagerie of centres hides inside.
Side lengths sort the shapes Euclid already named in Book One of the Elements: isosceles (two equal sides), equilateral (all equal), scalene (all different). Angles sort them again—right (one 90° corner), acute (all under 90°), obtuse (one over 90°). Gables and pediments often look isosceles; yield signs favour equilateral faces. The Great Pyramid of Giza’s faces are sometimes called equilateral, yet careful measurement finds them isosceles. Flags of Saint Lucia and the Philippines borrow the form; molecular geometry and deltahedra (polyhedra tiled by equilateral triangles) do too.
Special lines meet at celebrated points. Perpendicular bisectors concur at the circumcenter, centre of the unique circle through all three vertices; Thales’ theorem ties a right angle to a circumcenter on a side. Altitudes meet at the orthocenter, inside only for acute triangles. Angle bisectors meet at the incenter, heart of the largest inscribed circle. Medians meet at the centroid—also the centre of mass for a uniform triangular sheet—and each median is cut 2:1, longer toward the vertex. The nine-point circle, half the circumradius, touches the incircle at the Feuerbach point; orthocenter, nine-point centre, centroid, and circumcenter align on Euler’s line.
An exterior angle equals the sum of the two remote interior angles; the three exterior angles of any triangle total 360°, as for every convex polygon. Similar triangles share angle measures, so corresponding sides stay in proportion—AA, SAS with included angle, or SSS of proportional sides each suffice. Trigonometric sine, cosine, and tangent begin as side ratios in right triangles; the laws of sines and cosines extend the toolkit to scalene cases.
Beyond the flat plane, spherical and hyperbolic triangles live on curved surfaces, and geodesic triangles follow “straightest” paths on general surfaces. Four non-coplanar points in space span a tetrahedron—the triangle’s solid cousin.
Source: Triangle
