Triangle

I

Introduction

Triangle, geometric figure consisting of three points, called vertices, connected by three sides. In Euclidean plane geometry, the sides are straight line segments, as in figure 1. In spherical geometry, the sides are arcs of great circles, as in figure 2. The term triangle is sometimes used to describe a geometric figure having three vertices and sides that are arbitrary curves, as in figure 3.

II

Plane Triangles

A Euclidean plane triangle has three interior angles (the word interior is often omitted), each formed by two adjacent sides, as Ð CAB in figure 1, and six exterior angles, each formed by a side and the extension of an adjacent side, as Ð FEG in figure 4. A capital letter is customarily used to designate a vertex of a triangle, the angle at that vertex, or the measure of the angle in angular units; the corresponding lower case letter designates the side opposite the angle or its length in linear units. A side, or its length, is also designated by naming the endpoints; thus, in figure 1, the side opposite angle A is a or BC.

An angle A is acute if 0° < A < 90°; the angle is right if A = 90°; and it is obtuse if 90° < A < 180°. Because the sum of the angles of a triangle is 180°, a triangle can have at most one angle that is equal to or greater than 90°. A triangle is acute if all three of its angles are acute, as in figure 1; it is right if it has one right angle, as in figure 5; and it is obtuse if it has one obtuse angle, as in figure 4.

A triangle is scalene if no two of its sides are equal in length, as in figure 1; isosceles if two sides are equal, as in figure 6; and equilateral if all three sides are equal, as in figure 7. The side opposite the right angle in a right triangle, HK in figure 5, is called the hypotenuse; the other two sides are called legs. The two equal sides of an isosceles triangle are also called legs, and the included angle is called the vertex angle. The third side is called the base, and the adjacent angles are called the base angles.

If two sides of a triangle are of unequal size, the angles opposite are of unequal size; the larger side is opposite the larger angle. In the same way, if two angles are unequal, the sides opposite are unequal. Therefore, the three angles of a scalene triangle are of different sizes; the base angles of an isosceles triangle are equal; and the three angles of an equilateral triangle are equal. The side opposite a right angle or an obtuse angle is the longest side of the triangle. See Trigonometry: The General Triangle for the exact relation between the sizes of the angles and sides of a triangle.

In any triangle, an altitude is a line through a vertex perpendicular to the opposite side or its extension, for example, DX in figures 8a and 8b. A median is the line determined by a vertex and the midpoint of the opposite side, AN in figure 9. An interior angle bisector is the line through a vertex bisecting the interior angle at the vertex, AR in figure 10; similarly, an exterior angle bisector bisects the exterior angle at the vertex, AV in figure 10. A perpendicular bisector is a line perpendicular to a side at its midpoint, HK in figure 11. The terms altitude, median, and interior angle bisector are also often applied to the line segment determined by the vertex and the intersection with the opposite side or its extension.

The three altitudes of a triangle meet in a common point called the orthocentre, O in figures 8a and 8b. The three medians meet in a point called the centroid, M in figure 9. The centroid divides a median through it into two segments, and the segment having a vertex as endpoint is always twice as long as the other segment. The three angle bisectors meet in a point called the incentre (I in figure 10), and the three exterior angle bisectors meet in three points called the excentres, U, V, W of figure 10. The incentre and the three excentres are centres of circles tangent to the sides or side extensions of the triangle. The three perpendicular bisectors meet in a point called the circumcentre, H in figure 11, which is the centre of a circle passing through the three vertices.

If a, b, and c are the three sides of a triangle, and h_a is the altitude through vertex A, the area K is given by the formula K = yah_a. Many other formulas interrelate the various parts of a triangle.

III

Spherical Triangles

Many properties of plane triangles have analogues in spherical triangles, but the differences between the two types are important. For example, the sum of the angles of a spherical triangle is between 180° and 540° and varies with the size and shape of the triangle. A spherical triangle with one, two, or three right angles is called a rectangular, birectangular, or trirectangular triangle. A spherical triangle in which one, two, or three sides are quadrants (quarter circumferences of the sphere) is called a quadrantal, biquadrantal, or triquadrantal triangle.