Triangle
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. A triangle is a 2-simplex (see polytope).
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
Two triangles are said to be similar if one can be gotten by uniformly expanding the other. This is the case if and only if their corresponding angles are equal, and it occurs for example when two triangles share an angle and the sides opposite to that angle are parallel. The crucial fact about similar triangles is that 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.
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 angles A, B and C and sides a, b and c. The side a is opposite to the vertex at A and analogously for the other sides.
The sum of the angles A + B + C 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 plane right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If C is the right angle, we can write this as
- c2 = a2 + b2.
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:
- c2 = a2 + b2 - 2ab cos(C)
which is valid for all triangles, even if C 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.
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; this point is the center of the circumcircle, the circle passing through all three vertices. The radius of this circle is given by a/sin(A). As a consequence, we have the law of sines:
- sin(A)/a = sin(B)/b = sin(C)/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. Thales' theorem is also a consequence of the circumcircle formula. It states that if the center of the circumcircle is located on one side of the triangle, then the opposite angle is a right one.
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 touching all three sides.
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. 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 area S of a triangle can be computed in several ways:
- S = 1/2 × base × altitude
where you can choose any altitude.
- S = √(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 || AB × AC ||
where AB and AC are the vectors pointing from A to B respectively C, and ||AB × AC|| denotes the length of their cross-product (see vector).
If the vertex A is located at the origin 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
| x1 x2 |
| y1 y2 |
i.e.
- S = 1/2 × | x1y2 -y1x2 |
The triangle is also a musical instrument in the percussion family, usually made of steel and in the shape of a triangle. The triangle has a high-pitched, ringing tone.