Encyclopedia
A
triangle is one of the basic
shapes of
geometry: a
polygon with three vertices and three sides which are straight
line segments.
Any three non-
collinear points determine a triangle and a unique plane, i.e. two dimensional Cartesian space in
Euclidean geometry .
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 equilateral triangle is also equiangular, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon
- In an isosceles triangle at least two sides are of equal length. An isosceles triangle also has two equal internal angles . 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. 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 . The side opposite to 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 acute triangle has internal angles that are all smaller than 90° .
Basic facts
Elementary facts about triangles were presented by
Euclid in books 1-4 of his
Elements around 300 BCE.
A triangle is a
polygon and a 2-
simplex . All triangles are two-
dimensional.
Two triangles are said to be
similar if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional. 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 a, ß and ? and sides
a,
b and
c. The side
a is opposite to the vertex
A and angle a and analogously for the other sides.
In Euclidean geometry, the sum of the internal angles a + ß + ? is equal to two right angles . 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 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 ? 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.
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 : . The "30-60-90 triangle" has sides in the ratio of .
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. Often they are constructed by finding three lines associated in a symmetrical way with the three sides and then proving that the three lines meet in a single point: an important tool for proving the existence of these is
Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are
collinear: here
Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained.
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 the opposite side. This opposite side is called the
base of the altitude, and the point where the altitude intersects the base 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 acute.
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
incenter, 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, and divides the triangle into two equal areas. 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. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the
orthocenter. The radius of the nine-point circle 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 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.
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 and
h is the perpendicular distance between the base and the vertex not on the base. This can be shown with the following geometric construction.
To find the area of a given triangle , first make an exact copy of the triangle , 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.
The product of the
inradius and the
semiperimeter of a triangle also gives its area.
Using vectors
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
The altitude of a triangle can be found through an application of
trigonometry. Using the labelling as in the image on the left, the altitude is
h =
a sin ?. Substituting this in the formula
S = ½
bh derived above, the area of the triangle can be expressed as
S = ½
ab sin ?.
It is of course no coincidence that the area of a parallelogram is
ab sin ?.
Using coordinates
If vertex A is located at the origin of a
Cartesian coordinate system and the coordinates of the other two vertices are given by B = and C = , 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 is the 'Pythagorean' sum of the areas of the respective projections on the three principal planes :
Using Heron's formula
Yet another way to compute
S is
Heron's Formula:
where
s = ½ 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 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 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 will have its angles add up to less than 180° while a triangle on a positively curved figure will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, one would find that its angles were greater than 180°.
References
External links
- - solves for remaining sides and angles when given three sides or angles, supports degrees and radians.
- A triangle with three equilateral triangles. A purely geometric proof. It uses the Fermat point to prove Napoleon's theorem without transformations by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
- William Kahan: .
- Clark Kimberling: . Lists some 1600 interesting points associated with any triangle.
- Christian Obrecht: . Software package for creating illustrations of facts about triangles and other theorems in Euclidean geometry.
- at cut-the-knot
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- Analytical Geometry of Triangles
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- with interactive applets that are also useful in a classroom setting. Math Open Reference
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