Triangle

A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides of a triangle are given special names in the case of a right triangle, with the side opposite the right angle being termed the hypotenuse and the other two sides being known as the legs. All triangles are convex and bicentric. That portion of the plane enclosed by the triangle is called the triangle interior, while the remainder is the exterior.

The study of triangles is sometimes known as triangle geometry, and is a rich area of geometry filled with beautiful results and unexpected connections. In 1816, while studying the Brocard points of a triangle, Crelle exclaimed, "It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be?" (Wells 1991, p. 21).

It is common to label the vertices of a triangle in counterclockwise order as either , , (or , , ). The vertex angles are then given the same symbols as the vertices themselves. The symbols , , (or , , ) are also sometimes used (e.g., Johnson 1929), but this convention results in unnecessary confusion with the common notation for trilinear coordinates , and so is not recommended. The sides opposite the angles , , and (or , , ) are then labeled , , (or , , ), with these symbols also indicating the lengths of the sides (just as the symbols at the vertices indicate the vertices themselves as well as the vertex angles, depending on context).

An triangle is said to be acute if all three of its angles are all acute, a triangle having an obtuse angle is called an obtuse triangle, and a triangle with a right angle is called right. A triangle with all sides equal is called equilateral, a triangle with two sides equal is called isosceles, and a triangle with all sides a different length is called scalene. A triangle can be simultaneously right and isosceles, in which case it is known as an isosceles right triangle.

The semiperimeter of a triangle is defined as half its perimeter,

The area of a triangle can given by Heron's formula

There are also many other formulas for the triangle area.

The definition of the semiperimeter leads to the definitions

where is the inradius. A similar set of relations hold for Conway triangle notation , , , and .

The sum of angles in a triangle is radians (at least in Euclidean geometry; this statement does not hold in non-Euclidean geometry). This can be established as follows. Let ( be parallel to ) in the above diagram, then the angles and satisfy and , as indicated. Adding , it follows that

since the sum of angles for the line segment must equal two right angles. Therefore, the sum of angles in the triangle is also .

If a line is drawn parallel to one side of a triangle so that it intersects the other two sides, it divides them proportionally, i.e.,

(Jurgensen 1963, p. 251). In other words, a line parallel to a side of a triangle cutting the other two sides creates a triangle similar to the first.

Allowable side lengths , , and for a triangle are given by the set of inequalities , , , and , , , a statement that encapsulated in the so-called triangle inequality. The angles and sides of a triangle also satisfy an array of other beautiful triangle inequalities.

Specifying two angles and and a side uniquely determines a triangle with area

(the AAS theorem). Specifying an angle , a side , and an angle uniquely specifies a triangle with area

(the ASA theorem). Given a triangle with two sides, the smaller and the larger, and one known angle , acute and opposite , if , there are two possible triangles. If , there is one possible triangle. If , there are no possible triangles. This is the ASS theorem. Let be the base length and be the height. Then

(the SAS theorem). Finally, if all three sides are specified, a unique triangle is determined with area given by Heron's formula or by

where is the circumradius. This is the SSS theorem.

In triangle geometry, it is frequently very convenient to use a triple of coordinates defined relative to the distances from each side of a given so-called reference triangle. One form of such coordinates is known as trilinear coordinates , with all coordinates having the same sign corresponding to the triangle interior, one coordinate zero corresponding to a point on a side, two coordinates zero corresponding to a vertex, and coordinates having different signs corresponding to the triangle exterior.

The straightedge and compass construction of the triangle can be accomplished as follows. In the above figure, take as a radius and draw . Then bisect and construct . Extending to locate then gives the equilateral triangle . Another construction proceeds by drawing a circle of the desired radius centered at a point . Choose a point on the circle's circumference and draw another circle of radius centered at . The two circles intersect at two points, and , and is the second point at which the line intersects the first circle.

In Proposition IV.4 of the Elements, Euclid showed how to inscribe a circle (the incircle) in a given triangle by locating the incenter as the point of intersection of angle bisectors. In Proposition IV.5, he showed how to circumscribe a circle (the circumcircle) about a given triangle by locating the circumcenter as the point of intersection of the perpendicular bisectors. Unlike a general polygon with sides, a triangle always has both a circumcircle and an incircle. such polygons are called bicentric polygons.

A triangle with sides , , and can be constructed by selecting vertices (0, 0), , and , then solving

simultaneously to obtain

The angles of a triangle satisfy the law of cosines

as well as

where is the area (Johnson 1929, p. 11, with missing squared symbol added). The latter gives the pretty identity

In addition,

(F.J. n.d., p. 206; Borchardt and Perrott 1930) and

(Siddons and Hughes 1929), and

Additional formulas include

and

for even (Weisstein, Jan. 31, 2003 and Mar. 3, 2004).

Trigonometric functions of half angles in a triangle can be expressed in terms of the triangle sides as

where is the semiperimeter.

Let stand for a triangle side and for an angle, and let a set of s and s be concatenated such that adjacent letters correspond to adjacent sides and angles in a triangle. Triangles are uniquely determined by specifying three sides (SSS theorem), two angles and a side (AAS theorem), or two sides with an adjacent angle (SAS theorem). In each of these cases, the unknown three quantities (there are three sides and three angles total) can be uniquely determined. Other combinations of sides and angles do not uniquely determine a triangle: three angles specify a triangle only modulo a scale size (AAA theorem), and one angle and two sides not containing it may specify one, two, or no triangles (ASS theorem).

Dividing the sides of a triangle in a constant ratio and then drawing lines parallel to the adjacent sides passing through each of these points gives line segments which intersect each other and one of the medians in three places. If , then the extensions of the side parallels intersect the extensions of the medians.

The medians bisect the area of a triangle, as do the side parallels with ratio . The envelope of the lines which bisect the area a triangle forms three hyperbolic arcs. The envelope is somewhat more complicated, however, for lines dividing the area of a triangle into a constant but unequal ratio (Dunn and Petty 1972, Ball 1980, Wells 1991).

There are four circles which are tangent to the sides of a triangle, one internal (the incircle) and the rest external (the excircles). Their centers are the points of intersection of the angle bisectors of the triangle.

Any triangle can be positioned such that its shadow under an orthogonal projection is equilateral.