TRIANGLE (Lat. triangulus, three-cor nered, from tres, three ± angulus, angle). A figure formed by three intersecting lines. The sides of a plane triangle are straight lines, and those of a spherical triangle are geodetic lines or arcs of great circles. Triangles are called equi lateral, isosceles, and scalene according as three sides, two sides, or no sides are equal. Of two angles of a plane triangle that opposite the greater side is the greater, but the angles do not vary as the sides, the ratio of two sides being equal to that of the sines of the opposite angles. (See TRIGONOMETRY.) In a spherical triangle the sines of the sides are proportional to the sines of the opposite angles. The sum of the angles of a plane triangle is 180° according to Euclidean geometry (see GEOMETRY) ; but in the case of a spherical triangle the sum varies from 0° to 540°. The geometry of the triangle is extensive, and a few of the most important propositions are given under CONCURRENCE AND COLLINEARITY, and MAXIMA AND MINIMA. The area of any plane triangle is given by the for mulas mulas = and A = (s—a) (s—b) (s—c), where b is the base, /i the altitude, a, b, c the sides, and s the semi-perimeter. In case
the triangle is equilateral, a'2 = 4 , where a is the side. If one angle of a plane triangle is a right angle the triangle is called a triangle, the side opposite the right angle being the hypotenuse. The spherical triangle of one, two, or three right angles is called a rectangular, or trirectanvular triangle, re spectively. The spherical triangle in which one, two., or three sides are quadrants is called a quadrantal, biquadrantal, or triquadrantal tri angle, respectively. The area of a spherical tri angle is given by the formula A = where R is the radius of the sphere and E the spherical excess (q.v.), or A + C— ISO°. Consult Catalan, "Quelques formules relatives aux triangles rectilignes." in the Me/ noires con ronnes of the Brussels Academy (1S91) ; Casey, Sequel to Euclid (5th ed., Dublin. 1SSS).