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. The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points A triangle with vertices A, B, and C is denoted ABC.

In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. e. two-dimensional Cartesian space).

A triangle.

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. Properties The area of an equilateral triangle with sides of length a\\! An equilateral triangle is also an equiangular polygon, i. In Euclidean geometry, an equiangular polygon is a Polygon whose vertex angles are equal e. all its internal angles are equal—namely, 60°; it is a regular polygon. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called General properties These properties apply to both convex and star regular polygons [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

Triangles can also be classified according to their internal angles, described below using degrees of arc:

• A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one 90° internal angle (a right angle). This article describes the unit of angle For other meanings see Degree. Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. A hypotenuse is the longest side of a Right triangle, the side opposite of the Right angle. 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). In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called
• An acute triangle is an oblique triangle with internal angles all smaller than 90° (three acute angles). In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called 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. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek A triangle is a polygon and a 2-simplex (see polytope). In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit In Geometry, a simplex (plural simplexes or simplices) or n -simplex is an n -dimensional analogue of a triangle In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations All triangles are two-dimensional. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

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. 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 Like all convex polygons, the exterior angles of a triangle add up to 360 degrees.

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 Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater (In the special case of equality, two of the angles have collapsed to size zero, and the triangle has degenerated to a line segment. )

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 article is about proportionality the mathematical relation This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel.

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. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

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. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry A hypotenuse is the longest side of a Right triangle, the side opposite of the Right angle. If the hypotenuse has length c, and the legs have lengths a and b, then the theorem states that

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

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. A pair of Angles is complementary if the sum of their measures add up to 90 degrees.
• 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. In Trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general The law of sines ( sines law sine formula sine rule) in Trigonometry, is a statement about any Triangle in a 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. Ceva's theorem is a well-known theorem in elementary Geometry. In Geometry, three or more lines are said to be concurrent if they Intersect at a single point. 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. Menelaus' theorem, attributed to Menelaus of Alexandria, is a theorem about Triangles in Plane geometry. In this section just a few of the most commonly-encountered constructions are explained.

The circumcenter is the center of a circle passing through the three vertices of the triangle. In Geometry, the circumscribed circle or circumcircle of a Polygon is a Circle which passes through all the vertices of the polygon

A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i. In Geometry, bisection is the division of something into two equal or Congruent parts usually by a line, which is then called a bisector 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. In Geometry, the circumscribed circle or circumcircle of a Polygon is a Circle which passes through all the vertices of the polygon In Geometry, the circumscribed circle or circumcircle of a Polygon is a Circle which passes through all the vertices of the polygon Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the The diameter of this circle can be found from the law of sines stated above.

Thales' theorem implies that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. 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 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.

The intersection of the altitudes is the orthocenter. In Geometry, an altitude of a triangle is a Straight line through a vertex and Perpendicular to (i

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 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. In Geometry, an altitude of a triangle is a Straight line through a vertex and Perpendicular to (i 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. In Geometry, an orthocentric system is a set of four points in the plane one of which is the Orthocenter of the triangle

The intersection of the angle bisectors finds the center of the incircle. In Geometry, the incircle or inscribed circle of a triangle is the largest Circle contained in the triangle it touches (is Tangent

An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. In Geometry, bisection is the division of something into two equal or Congruent parts usually by a line, which is then called a bisector The three angle bisectors intersect in a single point, the incenter, the center of the triangle's incircle. In Geometry, the incircle or inscribed circle of a triangle is the largest Circle contained in the triangle it touches (is Tangent In Geometry, the incircle or inscribed circle of a triangle is the largest Circle contained in the triangle it touches (is Tangent 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. In Geometry, the incircle or inscribed circle of a triangle is the largest Circle contained in the triangle it touches (is Tangent The centers of the in- and excircles form an orthocentric system. In Geometry, an orthocentric system is a set of four points in the plane one of which is the Orthocenter of the triangle

The barycenter is the center of gravity. In Geometry, the centroid or barycenter of an object X in n- Dimensional space is the intersection of all Hyperplanes

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. In Geometry, a median of a Triangle is a Line segment joining a vertex to the Midpoint of the opposing side The three medians intersect in a single point, the triangle's centroid. In Geometry, the centroid or barycenter of an object X in n- Dimensional space is the intersection of all Hyperplanes 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.

Nine-point circle demonstrates a symmetry where six points lie on the edge of the triangle. In Geometry, the nine-point circle is a Circle that can be constructed for any given 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. In Geometry, the nine-point circle is a Circle that can be constructed for any given triangle. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. In Geometry, an altitude of a triangle is a Straight line through a vertex and Perpendicular to (i 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. 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

Euler's line is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red). In Geometry, the Euler line, named after Leonhard Euler, is a line determined from any Triangle that is not equilateral; it passes

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). In Geometry, the Euler line, named after Leonhard Euler, is a line determined from any Triangle that is not equilateral; it passes 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. Symmedians are three particular geometrical lines associated with every triangle. The three symmedians intersect in a single point, the symmedian point of the triangle. Symmedians are three particular geometrical lines associated with every triangle.

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. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which |AB × AC| is 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|.

The area of triangle ABC can also be expressed in terms of dot products as follows:

$\frac{1}{2} \sqrt{(\mathbf{AB} \cdot \mathbf{AB})(\mathbf{AC} \cdot \mathbf{AC}) -(\mathbf{AB} \cdot \mathbf{AC})^2} =\frac{1}{2} \sqrt{ |\mathbf{AB}|^2 |\mathbf{AC}|^2 -(\mathbf{AB} \cdot \mathbf{AC})^2} \, .$
Applying trigonometry to find the altitude h. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R

Using trigonometry

The height of a triangle can be found through an application of trigonometry. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. 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:

$S = \frac{1}{2}ab\sin \gamma = \frac{1}{2}bc\sin \alpha = \frac{1}{2}ca\sin \beta.$

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

$S = \frac{1}{2}ab\sin (\alpha+\beta) = \frac{1}{2}bc\sin (\beta+\gamma) = \frac{1}{2}ca\sin (\gamma+\alpha).$

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

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

For three general vertices, the equation is:

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

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. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the absolute value (or modulus) 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 n × n In Mathematics, Pythagorean addition is the following Binary operation: a \oplus b = \sqrt{a^2+b^2} e. x = 0, y = 0 and z = 0):

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

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:

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

where s = ½ (a + b + c) is the semiperimeter, or half of the triangle's perimeter. In Geometry, Heron's (or Hero's formula states that the Area (A of a Triangle whose sides have lengths a, b, and

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. Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere. In Mathematics, the term hyperbolic triangle has more than one meaning In

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°.

References

1. ^ Eric W. Weisstein, Equilateral triangle at MathWorld. This list of Triangle topics includes things related to the geometric shape either abstractly as in idealizations studied by geometers or in triangular arrays such as Pascal's triangle 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" (or Pythagorean Triangles The former are characterised In Geometry, the first Fermat point, or simply the Fermat point, also called Torricelli point, is the solution to the problem of finding a point F inside In Geometry, Pedoe's inequality, named after Daniel Pedoe, states that if a, b, and c are the lengths of the sides of a Triangle In Mathematics, Ono's inequality is a Theorem about Triangles in the Euclidean plane. In Euclidean plane Geometry, Lester's theorem, named after June Lester, states that in any Scalene triangle, the two Fermat points In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry The law of sines ( sines law sine formula sine rule) in Trigonometry, is a statement about any Triangle in a plane In Trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general In Trigonometry, the law of tangents is a statement about the relationship between the lengths of the three sides of a triangle and the tangents of the angles In Mathematics, Topology generalizes the notion of triangulation in a natural way as follows A triangulation of a Topological space X A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be A triangulated category is a mathematical category satisfying some Axioms that are based on the properties of the Homotopy category of spectra The inertia tensor \mathbf{J} of a Triangle (like the inertia tensor of any body can be expressed in terms of covariance \mathbf{C} of the body Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
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. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
3. ^ Eric W. Weisstein, Scalene triangle at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
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 MathWorld is an online Mathematics reference work created and largely written by Eric W