Triangle
- For alternate meanings, such as the musical instrument, see triangle (disambiguation).
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
\nA triangle can be characterized by whether any four of its elements (vertices, and/or elements of its sides) are plane to each other. If so, the triangle is called plane; in distinction, for instance, to a spherical or a hyperbolic triangle.
Plane triangles can be further classified according to their side lengths.\nThese classifications are as follows.\n* In an equilateral triangle all sides are equally long. An equilateral triangle is also equiangular, i.e. all its internal angles are equal—namely, 60°.\n* In an isosceles triangle two sides are equally long. An isosceles triangle also has two equal internal angles.\n* In a scalene triangle all sides have different lengths. The internal angles in a scalene triangle are all different.
\n\n | \n | \n | \n
\n\n| Equilateral | Isosceles | Scalene | \n
\n
Triangles can also be classified according to the size of their largest internal angle, described below using degrees of arc.\n* A
right triangle has one 90° internal angle (a
right angle). The side opposite the right angle is the
hypotenuse; it is the longest side in the right triangle. The other two sides are the
legs of the triangle.\n* An
obtuse triangle has one internal angle larger than 90° (an
obtuse angle).\n* An
acute triangle has internal angles that are all smaller than 90° (three acute angles).
\n\n | \n | \n | \n
\n\n| Right | Obtuse | Acute | \n
\n
Basic facts
Elementary facts about triangles were presented by Euclid in books 1-4 of his
Elements in around 300
BCE.
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 corresponding sides are in equal
proportion. 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 remainder 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.
\n |
\n| A triangle with vertices, sides and angles labelled |
\n
In Euclidean geometry, the sum of the angles α + β + γ is equal to two right angles (180° or π radians). This allows determination of the third angle of any triangle as soon as two angles are known.
\n |
\n| The Pythagorean theorem |
\n
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 vertex C is the right angle, 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.\nThe Pythagorean theorem can be generalized to the
law of cosines:
-
which is valid for all triangles, even if γ is not a right angle. \nThe 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 law of sines 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
\n |
\n| Circumcenter |
\n
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 can be found from the law of sines stated above.
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.
\n |
\n| Orthocenter |
\n
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.\nThe three vertices together with the orthocenter are said to form an
orthocentric system.
\n |
\n| Incircle |
\n
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, the center of the triangle's
incircle. The incircle is 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.
\n |
\n| Centroid |
\n
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, or on any line through the centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice as large as the distance between the centroid and the midpoint of the opposite side.
\n |
\n| Nine point circle |
\n
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 (at the
Feuerbach point) and the three excircles.
\n |
\n| Euler line |
\n
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). 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.
The center of the incircle is not in general located on Euler's line.
If one reflects a median at the angle bisector that passes through the same vertex, one obtains a
symmedian. The three symmedians intersect in a single point, the
symmedian point of the triangle.
Computing the area of a triangle
Calculating the area of a triangle is an elementary problem encountered often in many different situations. Various approaches exist, depending on what is known about the triangle. What follows is a selection of frequently used formulae for the area of a triangle.
Using geometry
The area S of a triangle is
S = ½
bh, 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. This can be shown with the following geometric construction.
\n |
\n\n\nThe triangle is first transformed into a parallelogram
\nwith twice the area of the triangle, then into a rectangle.\n | \n
\n
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. Cut off a part and join it at the other side of the parallelogram to form a rectangle. Because the area of the rectangle is
bh, the area of the given triangle must be ½
bh.
Using vectors
\n |
\n\n\nThe area of the parallelogram is the
\ncross product of the two vectors. \n | \n
\n
The area of a parallelogram can also be calculated by the use of
vectors. If
AB and
AC are vectors pointing from A to B and from A to C, respectively, the area of parallelogram ABDC is |
AB ×
AC|, the magnitude of the
cross product of vectors
AB and
AC. |
AB ×
AC| is also equal to |
h ×
AC|, where
h represents the altitude
h as a vector.
The area of triangle ABC is half of this, or
S = ½|
AB ×
AC|.
Using trigonometry
\n |
\n\n\nApplying trigonometry to
find the altitude h.\n | \n
\n
The altitude of a triangle can be found through an application of
trigonometry. Using the labelling as in the image on the right, the altitude is
h =
a sin γ. Substituting this in the formula
S = ½
bh derived above, the area of the triangle can be expressed as
S = ½
ab sin γ.
It is of course no coincidence that the area of a parallelogram is
ab sin γ.
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 = (
x1,
y1) and C = (
x2,
y2), then the area
S can be computed as 1/2 times the
absolute value of the
determinant
-
or
S = ½ |
x1y2 −
x2y1|.
Using Heron's formula
Yet another way to compute S is Heron's formula:\n:
where
s = ½ (
a +
b +
c) is the
semiperimeter, or one half of the triangle's perimeter.
See also
External links
Category:PolygonsCategory:Wikipedia Featured Articles\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n