- 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
A 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. These classifications are as follows.
- 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°.
- In an isosceles triangle two sides are equally long. An isosceles triangle also has two equal internal
angles.
- In a scalene triangle all sides have different lengths. The internal angles in a scalene triangle are all
different.
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| Equilateral |
Isosceles |
Scalene |
Triangles can also be classified according to the size of their largest internal angle, described below using degrees of arc.
- 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.
- An obtuse triangle has one internal angle larger than 90° (an obtuse angle).
- An acute triangle has internal angles that are all smaller than 90° (three acute angles).
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.
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| A triangle with vertices, sides and angles labelled |
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.
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
- 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 -
2abcosγ
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 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
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.
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, 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.
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.
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.
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 = 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. This can be
shown with the following geometric construction.
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The triangle is first transformed into a parallelogram
with twice the area of the triangle, then into a rectangle.
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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 ½bh.
Using vectors
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The area of the parallelogram is the
cross product of the two vectors.
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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 |h × AC|. Here |h × AC| represents the
altitude h and the length of the cross product vector
AC.
The area of triangle ABC is half of this, or S = ½|h × AC|.
Using trigonometry
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Applying trigonometry to
find the altitude h.
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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 = ½bh
derived above, the area of the triangle can be expressed as S = ½ab 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 − x2y
1|.
Using Heron's formula
Yet another way to compute S is Heron's formula:
-
where s = ½ (a + b + c) is the
semiperimeter, or one half of the triangle's perimeter.
See also
External links
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