Geometry

Triangles are plane figures determined by three non-collinear points called "vertices." They are made up of the segments, called sides, which join them. Although the sides are segments rather than rays, each pair of them makes up one of the triangle's angles.

Triangles may be classified by the size of their angles or by the lengths of their sides. Triangles whose angles are all less than right angles are called "acute." Those with one right angle are "right" triangles. Those with one angle larger than a right angle are "obtuse." (In a right triangle the side opposite the right angle is called the "hy potenuse" and the other two sides "legs.") Triangles with no equal sides are "scalene" triangles. Those with two equal sides are "isosceles." Those with three equal sides are "equilateral." There is no direct connection between the size of the angles of a triangle and the lengths of its sides. The longest side, however, will be opposite the largest angle; and the shortest side, opposite the smallest angle. Equal sides will be opposite equal angles.

In comparing triangles it is useful to set up a correspondence between them and to name corresponding vertices in the same order. If CXY and PST are two such triangles, then angles C and P correspond; sides CY and PT correspond; and so on.

Two triangles are "congruent" when their six corresponding parts are equal. Congruent triangles have the same size and shape, although one may be the mirror image of the other. Triangles ABC and FDE are congruent provided that the sides and angles which appear to be equal are in fact equal.

One can show that two triangles are congruent without establishing the equality of all six parts. Two triangles will be congruent whenever

1. Two sides and the included angle of one are equal to two sides and the included angle of the other (SAS congruence).
2. Two angles and the included side of one are equal to two angles and the included side of the other (ASA congruence).
3. Three sides of one are equal to three sides of the other (SSS congruence).

Triangle congruence applies not only to two different triangles. It also applies to one triangle at two different times or to one triangle looked at in two different ways. For example, when the girders of a bridge are strengthened with triangular braces, each triangle stays congruent to itself over a period of time, and does so by virtue of SSS congruence.

Two triangles can also be similar. Similar triangles have the same shape, but not necessarily the same size. They are alike in the way that a snapshot and an enlargement of it are alike. When two triangles are similar, corresponding angles are equal and corresponding sides are proportional.

One can show that two triangles are similar without showing that all the angles are equal and all the sides proportional. Two triangles will be similar when

1. Two sides of one triangle are proportional to two sides of another triangle and the included angles are equal (SAS similarity).
2. Two angles of one triangle are equal to two angles of another triangle (AA similarity).
3. Three sides of one triangle are proportional to three sides of another triangle (SSS similarity).

The properties of similar triangles are widely used. Artists, for example, use them in making smaller or larger versions of a picture. Map makers use them in drawing maps; and users, in reading them.

Figure 3 shows a right triangle in which an altitude BD has been drawn to the hypotenuse AC. By AA similarity, the triangles ABC, ADB, and BDC are similar to one another.

By virtue of these similarities one can write AC/BC = BC/DC and AC/AB = AB/AD. Then, using AD + DC = AC and a little algebra, one ends up with (AB)² + (BC)² = (AC)², or the Pythagorean theorem: "In a right triangle the sum of the squares on the legs is equal to the square on the hypotenuse." This neat proof was discovered by Bhaskara, mentioned earlier.

The altitude BD in Figure 3 is also the mean proportional between AD and DC. That is, AD/BD = BD/DC.

In triangle ABC, if DE is a line drawn parallel to AC, it creates a triangle similar to ABC. It therefore divides AB and BC proportionally. Conversely, a line which divides two sides of a triangle proportionally is parallel to the third side. A special case of this is a segment joining the midpoints of two sides of a triangle. It is parallel to the third side and half its length.

Each triangle has four sets of lines associated with it: medians, altitudes, angle bisectors, and perpendicular bisectors of the sides. In each set, the three lines are, remarkably, concurrent, that is, they all pass through a single point. In the case of the medians, which are lines from a vertex to the midpoint of the opposite side, the point of concurrency is the "centroid," the center of gravity. The angle bisectors are concurrent at the "incenter," the center of a circle tangent to the three sides. The perpendicular bisectors of the sides are concurrent at the "circumcenter," the center of a circle passing through all three vertices. The altitudes, which are lines from a vertex perpendicular to the opposite side, are concurrent at the "orthocenter."