triangle

A triangle is a planar region delimited by three lines, i.e. it is a polygon with three angles.

In Euclidean geometry, the angle sum of a triangle is always equal to $180^{\circ}$ . In the figure: $A+B+C=180^{\circ}$ .

In hyperbolic geometry, the angle sum of a triangle is always strictly positive and strictly less than $180^{\circ}$ . In the figure: $0^{\circ}<A+B+C<180^{\circ}$ .

In spherical geometry, the angle sum of a triangle is always strictly greater than $180^{\circ}$ and strictly less than $540^{\circ}$ . In the figure: $180^{\circ}<A+B+C<540^{\circ}$ .

Also in spherical geometry, a triangle has these additional requirements: It must be strictly contained in a hemisphere of the sphere that is serving as the model for spherical geometry, and all of its angles must have a measure strictly less that $180^{\circ}$ .

Triangles can be classified according to the number of their equal sides. So, a triangle with 3 equal sides is called equilateral (http://planetmath.org/RegularTriangle), a triangle with 2 equal sides is called isosceles, and finally a triangle with no equal sides is called scalene. Notice that an is also isosceles, but there are isosceles triangles that are not equilateral.

In Euclidean geometry, triangles can also be classified according to the of the greatest of its three (inner) angles. If the greatest of these is acute (and therefore all three are acute), the triangle is called an acute triangle. If the triangle has a right angle, it is a right triangle. If the triangle has an obtuse angle, it is an obtuse triangle.

Area of a triangle

There are several ways to a triangle’s area.

In hyperbolic and spherical , the area of a triangle is equal to its defect (measured in radians).

For the rest of this entry, only Euclidean geometry will be considered.

Many for the area of a triangle exist. The most basic one is $\displaystyle A=\frac{1}{2}bh$ , where $b$ is its base and $h$ is its height. Following is a of another for the area of a triangle.

Let $a, b, c$ be the sides and $A, B, C$ the interior angles to them. Let $h_{a},h_{b},h_{c}$ be the heights drawn upon $a, b, c$ respectively, $r$ the inradius and $R$ the circumradius. Finally, let $\displaystyle s=\frac{a+b+c}{2}$ be the semiperimeter. Then

Area	$\displaystyle=$	$\displaystyle\frac{ah_{a}}{2}=\frac{bh_{b}}{2}=\frac{ch_{c}}{2}$
	$\displaystyle=$	$\displaystyle\frac{ab\sin C}{2}=\frac{bc\sin A}{2}=\frac{ca\sin B}{2}$
	$\displaystyle=$	$\displaystyle\frac{abc}{4R}$
	$\displaystyle=$	$\displaystyle sr$
	$\displaystyle=$	$\displaystyle\sqrt{s(s-a)(s-b)(s-c)}$

The last is known as Heron’s formula.

With the coordinates of the vertices $(x_{1},\,y_{1})$ , $(x_{2},\,y_{2})$ , $(x_{3},\,y_{3})$ of the triangle, the area may be expressed as

\pm\frac{1}{2}\left|\begin{matrix}x_{1}&y_{1}&1\\ x_{2}&y_{2}&1\\ x_{3}&y_{3}&1\end{matrix}\right|

(cf. the volume of tetrahedron (http://planetmath.org/Tetrahedron)).

Inequalities for the area are Weizenbock’s inequality and the Hadwiger-Finsler inequality.

Angles in a triangle

1.

the sum of the angles in a triangle is $\pi$ radians ( $180^{\circ}$ )
2.

sines law
3.

cosines law
4.

Mollweide’s equations

Special geometric objects for a triangle

1.

incenter
2.

inscribed circle
3.

circumcenter
4.

circumscribed circle
5.

centroid
6.

orthocenter
7.

Lemoine point, Lemoine circle
8.

Gergonne point, Gergonne triangle
9.

orthic triangle
10.

pedal triangle
11.

medial triangle
12.

Euler Line

Title	triangle
Canonical name	Triangle
Date of creation	2013-03-22 11:43:51
Last modified on	2013-03-22 11:43:51
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	54
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 51-00
Classification	msc 51M05
Classification	msc 00A05
Classification	msc 51M10
Classification	msc 55-00
Classification	msc 55-01
Related topic	SinesLaw
Related topic	EulerLine
Related topic	Median
Related topic	PythagorasTheorem
Related topic	Hypotenuse
Related topic	Orthocenter
Related topic	OrthicTriangle
Related topic	IsoscelesTriangle
Related topic	CevasTheorem
Related topic	Cevian
Related topic	SinesLawProof
Related topic	FundamentalTheoremOnIsogonalLines
Related topic	Incenter
Related topic	EquilateralTriangle
Related topic	TrigonometricVersionOfCevasTheorem
Related topic	HeronsFo
Defines	acute triangle
Defines	right triangle
Defines	obtuse triangle