Triangle

This article is about the geometric shape; for the musical instrument, see triangle (instrument).

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A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments.

Types of triangles

A triangle is called

• equilateral if all its sides have the same length (or equivalently: all its angles are equal)
• isosceles if two of its sides have the same length (or equivalently: two of its angles are equal)
• scalene if all sides have different lengths
• right if one of its angles is a right angle (90 degrees or π/2 radians). The side opposite the right angle is called the hypotenuse. It is the longest side in the right triangle.
• obtuse if one angle is bigger than a right one
• acute if each angle is smaller than a right one

Basic facts

A triangle is a polygon and a 2-simplex (see polytope).

Two triangles are said to be similar if one can be produced by uniformly expanding the other. In this case, the lengths of their sides are proportional. That is, if the longest side of a triangle is twice that of the longest side of a similar triangle, say, then the shortest side will also be twice that of the shortest side of the other triangle, and the median side will be twice that of the other triangle. Also, the ratio of the longest side to the shortest in the first triangle will be the same as the ratio of the longest side to the shortest in the other triangle. The crucial fact is that two triangles are similar if and only if their corresponding angles are equal, and this occurs for example when two triangles share an angle and the sides opposite to that angle are parallel.

Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry.

In the sequel, we will consider a triangle with vertices A, B and C, angles α, β and γ and sides a, b and c. The side a is opposite to the vertex A and angle α and analogously for the other sides.

The sum of the angles α + β + γ is equal to two right angles (180 degrees or π radians). This allows to determine the third angle of any triangle as soon as two angles are known.

A central theorem is the Pythagorean theorem stating that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If γ is the right angle, we can write this as

$c^{2}=a^{2}+b^{2}$

This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the third -- something unique to right triangles. The Pythagorean theorem can be generalized to the law of cosines:

$c^{2}=a^{2}+b^{2}-2ab\cos(\gamma )$

which is valid for all triangles, even if γ is not a right angle. The law of cosines can be used to compute the side lengths and angles of a triangle as soon as all three sides or two sides and an enclosed angle are known.

The law of sines states

$\sin(\alpha )/a=\sin(\beta )/b=\sin(\gamma )/c$

which can be used to compute the side lengths for a triangle as soon as two angles and one side are known. If two sides and an unenclosed angle is known, the law of sines may also be used; however, in this case there may be zero, one or two solutions.

Points, lines and circles associated with a triangle

A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle is given by a/sin(α).

Thales' theorem states that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse.

An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is not obtuse. The three vertices together with the orthocenter are said to form an orthocentric system.

An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point; this point is the center of the triangle's incircle, the circle which lies inside the triangle and touches all three sides. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system.

A median of a triangle is a straight line through a vertex and the midpoint of the opposite side. The three medians intersect in a single point, the triangle's centroid. This is also the triangle's center of gravity: if the triangle were made out of wood, say, you could balance it on its centroid.

The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine point circle. Its radius is half that of the circumcircle. It touches the incircle and the three excircles.

The centroid, orthocenter, circumcenter and center of the nine point circle (but not necessarily the center of the incircle) all lie on a single line, known as Euler's line. The center of the nine point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.

Computing the area of a triangle

The area S of a triangle can be computed in several ways:

$S={1/2}\times {\mbox{base}}\times {\mbox{altitude}}$

where the altitute can be chosen arbitrarily,

$S={\sqrt {s(s-a)(s-b)(s-c)}}$

where s = 1/2 (a + b + c) is one half of the triangle's perimeter (Heron's formula),

$S=sr$

where s is defined as above and r is the radius of the triangle's incircle,

$S={1/2}\left|AB\times AC\right|$

where AB and AC are the vectors pointing from A to B respectively C, and |AB × AC| denotes the length of their cross product. This is because |AB × AC| represents the area of the parallelogram formed by these vectors, and thus the area of the triangle is half this.

If the vertex A is located at the origin (0,0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (x₁, y₁) and C = (x₂, y₂), then the area S can be computed as 1/2 times the absolute value of the determinant

${\begin{vmatrix}x_{1}&x_{2}\\y_{1}&y_{2}\end{vmatrix}}$

i.e.

$S={1/2}\left|x_{1}y_{2}-y_{1}x_{2}\right|$

See also

• triangular_number
• triangle (instrument)
• Pascal's triangle
• synthetic geometry

External links

• Clark Kimberling: Encyclopedia of triangle centers. Lists some 1600 interesting points associated with any triangle.