A musical instrument of great antiquity, made from a steel rod in the form of a triangle, with one corner left open. It is struck with a short metal beater to produce a high, silvery sound of indefinite pitch.
Any three non-collinear points determine a triangle and a unique plane, i.e.
From the systemics perspective, triangle is the structure of every system composed with three reciprocally connected/interrelated abstract or real objects.
Types of triangles
Triangles can be classified according to the relative lengths of their sides:
In an equilateral triangle all sides are of equal length. An isosceles triangle also has two equal internal angles (namely, the angles where each of the equal sides meets the third side). An equilateral triangle is actually also an isosceles triangle, but not all isosceles triangles are equilateral triangles In a scalene triangle all sides have different lengths.| Equilateral | Isosceles | Scalene |
Triangles can also be classified according to the size of their largest internal angle, described below using degrees of arc.
| Right | Obtuse | Acute |
Basic facts
Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 BCE.
Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other. 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.
In the remainder we will consider a triangle with vertices A, B and C, angles α, β and γ and sides a, b and c.
A central theorem is the Pythagorean theorem stating that in any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. If side C is the hypotenuse, we can write this as
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:
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
where d is the diameter of the circumcircle (the circle which passes through all three points of the triangle). The law of sines can be used to compute the side lengths for a triangle as soon as two angles and one side are known.
There are two special right triangles that appear commonly in geometry. The so-called "45-45-90 triangle" has angles with those angle measures and the ratio of its sides is : .
Points, lines and circles associated with a triangle
There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them.
A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e.
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. The orthocenter lies inside the triangle if and only if the triangle is acute.
An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half.
A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. 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, or on any line through the 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.
The centroid (yellow), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red point) all lie on a single line, known as Euler's line (red line).
Computing the area of a triangle
Calculating the area of a triangle is an elementary problem encountered often in many different situations.
Using geometry
The area S of a triangle is S = ½bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base.
The triangle is first transformed into a parallelogram with twice the area of the triangle, then into a rectangle.To find the area of a given triangle (green), first make an exact copy of the triangle (blue), rotate it 180°, and join it to the given triangle along one side to obtain a parallelogram.
The product of the inradius and the semiperimeter of a triangle also gives its area.
The area of triangle ABC is half of this, or S = ½|AB × AC|.
Using trigonometry
The altitude of a triangle can be found through an application of trigonometry.
Using coordinates
If 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 = (xB, yB) and C = (xC, yC), then the area S can be computed as ½ times the absolute value of the determinant
For three general vertices, the equation is:
In three dimensions, the area of a general triangle {A = (xA, yA, zA), B = (xB, yB, zB) and C = (xC, yC, zC)} is the 'Pythagorean' sum of the areas of the respective projections on the three principal planes (i.e. x=0, y=0 and z=0):
Using Heron's formula
Yet another way to compute S is Heron's Formula:
where s = ½ (a + b + c) is the semiperimeter, or half of the triangle's perimeter.
Non-planar triangles
A non-planar triangle is a triangle which is not contained in a (flat) plane. Examples of non-planar triangles in noneuclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.
While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°.
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