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Encyclopedia > Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC. Look up triangle in Wiktionary, the free dictionary. ... This article is about describing the shape of an object. ... For other uses, see Geometry (disambiguation). ... Look up polygon in Wiktionary, the free dictionary. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ...

In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. two-dimensional Cartesian space). Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... This article is about the mathematical construct. ... In mathematics, n-dimensional Euclidean space (also called Cartesian space or n-space) refers to the space of ordered n-tuples of real numbers along with the associated operations of component-wise addition and scalar multiplication which make it into a vector space, and the dot product which makes it...

A triangle.

Image File history File links Triangle_illustration. ... Image File history File links Triangle_illustration. ...

## 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 an equiangular polygon, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon.[1]
• In an isosceles triangle, two sides are of equal length (originally and conventionally limited to exactly two).[2] An isosceles triangle also has two equal angles: the angles opposite the two equal sides.
• In a scalene triangle, all sides have different lengths. The internal angles in a scalene triangle are all different.[3]
 Equilateral Isosceles Scalene

• A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one 90° internal angle (a right 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 or catheti (singular: cathetus) of the triangle.
• An oblique triangle has no internal angle equal to 90°.
• An obtuse triangle is an oblique triangle with one internal angle larger than 90° (an obtuse angle).
• An acute triangle is an oblique triangle with internal angles all smaller than 90° (three acute angles). An equilateral triangle is an acute triangle, but not all acute triangles are equilateral triangles.
 Right Obtuse Acute $underbrace{qquad qquad qquad qquad qquad qquad}_{}$ Oblique

## 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 (see polytope). All triangles are two-dimensional. For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... Look up polygon in Wiktionary, the free dictionary. ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions. ... 2-dimensional renderings (ie. ...

The angles of a triangle add up to 180 degrees. An exterior angle of a triangle (an angle that is adjacent and supplementary to an internal angle) is always equal to the two angles of a triangle that it is not adjacent/supplementary to. Like all convex polygons, the exterior angles of a triangle add up to 360 degrees. External angles law In geometry, an interior angle (or internal angle) is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon. ... Look up convex in Wiktionary, the free dictionary. ...

The sum of the lengths of any two sides of a triangle always exceeds the length of the third side. That is the triangle inequality. (In the special case of equality, two of the angles have collapsed to size zero, and the triangle has degenerated to a line segment.) In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...

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. Several equivalence relations in mathematics are called similarity. ... This article is about proportionality, the mathematical relation. ...

A few basic postulates and theorems about similar triangles:

• Two triangles are similar if at least two corresponding angles are equal.
• If two corresponding sides of two triangles are in proportion, and their included angles are equal, the triangles are similar.
• If three sides of two triangles are in proportion, the triangles are similar.

For two triangles to be congruent, each of their corresponding angles and sides must be equal (6 total). A few basic postulates and theorems about congruent triangles:

• SAS Postulate: If two sides and the included angles of two triangles are correspondingly equal, the two triangles are congruent.
• SSS Postulate: If every side of two triangles are correspondingly equal, the triangles are congruent.
• ASA Postulate: If two angles and the included sides of two triangles are correspondingly equal, the two triangles are congruent.
• AAS Theorem: If two angles and any side of two triangles are correspondingly equal, the two triangles are congruent.
• Hypotenuse-Leg Theorem: If the hypotenuses and one leg of two right triangles are correspondingly equal, the triangles are congruent.

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. Sine redirects here. ... This article is about angles in geometry. ... Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ...

In Euclidean geometry, the sum of the internal angles of a triangle is equal to 180°. This allows determination of the third angle of any triangle as soon as two angles are known.

The Pythagorean theorem

A central theorem is the Pythagorean theorem, which states in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c, and the legs have lengths a and b, then the theorem states that Image File history File links Pythagorean. ... Image File history File links Pythagorean. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... A right triangle and its hypotenuse, h, along with catheti, c1 and c2. ...

$a^2 + b^2=c^2. ,$

The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle is a right triangle.

Some other facts about right triangles:

• The acute angles of a right triangle are complementary.
• If the legs of a right triangle are equal, then the angles opposite the legs are equal, acute and complementary, and thus are both 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the length of a leg times the square root of two.
• In a 30-60 right triangle, in which the acute angles measure 30 and 60 degrees, the hypotenuse is twice the length of the shorter side.
• In all right triangles, the median on the hypotenuse is the half of the hypotenuse.

For all triangles, angles and sides are related by the law of cosines and law of sines. A pair of complementary angles, because they add up to 90 degrees. ... Fig. ... In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ...

## 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 (or vertices) 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. Cevas Theorem (pronounced Cheva) is a very popular theorem in elementary geometry. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... Menelaus theorem, case 1: line DEF passes inside triangle ABC Menelaus theorem, attributed to Menelaus of Alexandria, is a theorem about triangles in plane geometry. ...

The circumcenter is the center of a circle passing through the three vertices of the 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. Image File history File links Triangle. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ... For the bisection theorem, see ham sandwich theorem. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ... This article is about the shape and mathematical concept of circle. ...

Thales' theorem implies 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. In geometry, Thales theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. ...

The intersection of the altitudes is the orthocenter.

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 acute. The three vertices together with the orthocenter are said to form an orthocentric system. Image File history File links Triangle. ... In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i. ... In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i. ... In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i. ... In geometry, an orthocentric system is a set of four points in the plane where one point is the orthocenter of the triangle formed by the other three. ...

The intersection of the angle bisectors finds the center of the incircle.

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.
Image File history File links Triangle. ... It has been suggested that this article or section be merged into Stanford University. ... For the numerical analysis algorithm, see bisection method. ... In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. ... It has been suggested that this article or section be merged into Stanford University. ... In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. ... In geometry, an orthocentric system is a set of four points in the plane where one point is the orthocenter of the triangle formed by the other three. ...

The barycenter is the center of gravity.

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. Image File history File links Triangle. ... Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ... The triangle medians and the centroid. ... Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ... This article or section may contain original research or unverified claims. ...

Nine-point circle demonstrates a symmetry where six points lie on the edge of the triangle.

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 (at the Feuerbach point) and the three excircles.
Image File history File links Triangle. ... In geometry, the nine-point circle is a circle that can be constructed for any given triangle. ... In geometry, the nine-point circle is a circle that can be constructed for any given triangle. ... In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i. ... In geometry, the nine point circle is a circle that can be constructed for any given triangle. ... In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. ...

Euler's line is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).

The centroid (yellow), orthocenter (blue), circumcenter (green) and barycenter 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. Image File history File links Triangle. ... In geometry, Eulers line (red line in the image), named after Leonhard Euler, is the line passing through the orthocenter (blue), the circumcenter (green), the centroid (yellow), and the center of the nine-point circle (red point) of any triangle. ... In geometry, Eulers line (red line in the image), named after Leonhard Euler, is the line passing through the orthocenter (blue), the circumcenter (green), the centroid (yellow), and the center of the nine-point circle (red point) of any triangle. ...

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.
In geometry, three special lines are associated with every triangle, the triangles symmedians. ... In geometry, three special lines are associated with every triangle, the triangles symmedians. ...

## Computing the area of a triangle

Calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known, and simplest formula is

$S=frac{1}{2}bh$

where S is area, b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term 'base' denotes any side, and 'height' denotes the length of a perpendicular from the point opposite the side onto the side itself.

Although simple, this formula is only useful if the height can be readily found. For example, the surveyor of a triangular field measures the length of each side, and can find the area from his results without having to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle.[4]

### Using vectors

The area of a parallelogram can be calculated using vectors. Let vectors AB and AC point respectively from A to B and from A to C. The area of parallelogram ABDC is then |AB × AC|, which is the magnitude of the cross product of vectors AB and AC. |AB × AC| is equal to |h × AC|, where h represents the altitude h as a vector. This article is about vectors that have a particular relation to the spatial coordinates. ... For the cross product in algebraic topology, see KÃ¼nneth theorem. ...

The area of triangle ABC is half of this, or S = ½|AB × AC|.

The area of triangle ABC can also be expressed in terms of dot products as follows: In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...

$frac{1}{2} sqrt{(mathbf{AB} cdot mathbf{AB})(mathbf{AC} cdot mathbf{AC}) -(mathbf{AB} cdot mathbf{AC})^2} =frac{1}{2} sqrt{ |mathbf{AB}|^2 |mathbf{AC}|^2 -(mathbf{AB} cdot mathbf{AC})^2} , .$
Applying trigonometry to find the altitude h.

Image File history File links Triangle. ...

### Using trigonometry

The height 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 γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as: Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ...

$S = frac{1}{2}absin gamma = frac{1}{2}bcsin alpha = frac{1}{2}casin beta.$

Furthermore, since sin α = sin (π - α) = sin (β + γ), and similarly for the other two angles:

$S = frac{1}{2}absin (alpha+beta) = frac{1}{2}bcsin (beta+gamma) = frac{1}{2}casin (gamma+alpha).$

### 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 = (xByB) and C = (xCyC), then the area S can be computed as ½ times the absolute value of the determinant Fig. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

$S=frac{1}{2}left|detbegin{pmatrix}x_B & x_C y_B & y_C end{pmatrix}right| = frac{1}{2}|x_B y_C - x_C y_B|.$

For three general vertices, the equation is:

$S=frac{1}{2} left| detbegin{pmatrix}x_A & x_B & x_C y_A & y_B & y_C 1 & 1 & 1end{pmatrix} right| = frac{1}{2} big| x_A y_C - x_A y_B + x_B y_A - x_B y_C + x_C y_B - x_C y_A big|.$

In three dimensions, the area of a general triangle {A = (xAyAzA), B = (xByBzB) and C = (xCyCzC)} 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):

$S=frac{1}{2} sqrt{ left( detbegin{pmatrix} x_A & x_B & x_C y_A & y_B & y_C 1 & 1 & 1 end{pmatrix} right)^2 + left( detbegin{pmatrix} y_A & y_B & y_C z_A & z_B & z_C 1 & 1 & 1 end{pmatrix} right)^2 + left( detbegin{pmatrix} z_A & z_B & z_C x_A & x_B & x_C 1 & 1 & 1 end{pmatrix} right)^2 }.$

### Using Heron's formula

The shape of the triangle is determined by the lengths of the sides alone. Therefore the area S also can be derived from the lengths of the sides. By Heron's formula: A triangle with sides a, b, and c. ...

$S = sqrt{s(s-a)(s-b)(s-c)}$

where s = ½ (a + b + c) is the semiperimeter, or half of the triangle's perimeter.

An equivalent way of writing Heron's formula is

$S = frac{1}{4} sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}.$

## 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. Right spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... In mathematics, the term hyperbolic triangle has more than one meaning. ... Lines through a given point P and asymptotic to line l. ...

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°. Thus, if one were to draw a giant triangle on the surface of the Earth, one would find that the sum of its angles were greater than 180°.

A triangular number is the sum of the n natural numbers from 1 to n. ... Two types of special right triangles appear commonly in geometry, the angle based and the side based triangles. ... Construction for the fermat point. ... In mathematics, the Hadwiger-Finsler inequality is a result on the geometry of triangles in the Euclidean plane, named after the mathematicians Hugo Hadwiger and Paul Finsler. ... In geometry, Pedoes inequality, named after Dan Pedoe, states that if a, b, and c are the lengths of the sides of a triangle with area f, and A, B, and C are the lengths of the sides of a triangle with area F, then with equality if and... In mathematics, Onos inequality is a theorem about triangles in the Euclidean plane. ... An example of congruence. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ... Fig. ... In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane. ... Topology generalizes the notion of triangulation in a natural way as follows: A triangulation of a topological space is a simplicial complex , homeomorphic to , together with a homeomorphism . ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... A triangulated category is a mathematical category satisfying some axioms that are based on the properties of a derived category. ...

## References

1. ^ Eric W. Weisstein, Equilateral triangle at MathWorld.
2. ^ Mathematicians have traditionally followed Euclid (Book 1 definition 20) in defining an isosceles triangle as having exactly two sides equal, so that equilateral triangles are excluded; but modern references tend to include equilateral triangles: Wiktionary definition of isosceles triangle, Eric W. Weisstein, Isosceles triangle at MathWorld.
3. ^ Eric W. Weisstein, Scalene triangle at MathWorld.
4. ^ Eric W. Weisstein, Triangle area at MathWorld.

Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Results from FactBites:

 Area of a Triangle (520 words) For example, in the diagram to the left, the area of each triangle is equal to one-half the area of the parallelogram. However, depending on the triangle, the height may or may not be a side of the triangle. In the triangles in Examples 1 and 3, the lateral sides are not perpendicular to the base, so a dotted line is drawn to represent the height.
 Triangle (geometry) - MSN Encarta (834 words) A capital letter is customarily used to designate a vertex of a triangle, the angle at that vertex, or the measure of the angle in angular units; the corresponding lower case letter designates the side opposite the angle or its length in linear units. Therefore, the three angles of a scalene triangle are of different sizes; the base angles of an isosceles triangle are equal; and the three angles of an equilateral triangle are equal (an equilateral triangle is also equiangular). For example, the sum of the angles of a spherical triangle is between 180° and 540° and varies with the size and shape of the triangle.
More results at FactBites »

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