Changes

==Origin==

==Origin==

[https://nordan.daynal.org/wiki/index.php?title=English#ca._1100-1500_.09THE_MIDDLE_ENGLISH_PERIOD Middle English], from Anglo-French, from [[Latin]] ''triangulum'', from neuter of ''triangulus'' triangular, from ''tri''- + ''angulus'' [[angle]]

[https://nordan.daynal.org/wiki/index.php?title=English#ca._1100-1500_.09THE_MIDDLE_ENGLISH_PERIOD Middle English], from Anglo-French, from [[Latin]] ''triangulum'', from neuter of ''triangulus'' triangular, from ''tri''- + ''angulus'' [[angle]]

*[http://en.wikipedia.org/wiki/14th_century 14th Century]

*[https://en.wikipedia.org/wiki/14th_century 14th Century]

==Definitions==

==Definitions==

*1: a [http://en.wikipedia.org/wiki/Polygon polygon] having [[three]] sides — [[compare]] spherical triangle

*1: a [https://en.wikipedia.org/wiki/Polygon polygon] having [[three]] sides — [[compare]] spherical triangle

*2a : a percussion instrument consisting of a rod of steel bent into the form of a triangle open at one [[angle]] and sounded by striking with a small metal rod

*2a : a percussion instrument consisting of a rod of steel bent into the form of a triangle open at one [[angle]] and sounded by striking with a small metal rod

:b : a drafting instrument consisting of a thin flat right-angled triangle of wood or plastic with acute angles of 45 degrees or of 30 degrees and 60 degrees

:b : a drafting instrument consisting of a thin flat right-angled triangle of wood or plastic with acute angles of 45 degrees or of 30 degrees and 60 degrees

==Description==

==Description==

A '''triangle''' is one of the basic shapes of [[geometry]]: a [http://en.wikipedia.org/wiki/Polygon 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 [[File:Triangle.jpg‎]].

A '''triangle''' is one of the basic shapes of [[geometry]]: a [https://en.wikipedia.org/wiki/Polygon 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 [[File:Triangle.jpg‎]].

In [http://en.wikipedia.org/wiki/Euclidean_geometry Euclidean geometry] any three [[points]], when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional [http://en.wikipedia.org/wiki/Euclidean_space Euclidean space]).

In [https://en.wikipedia.org/wiki/Euclidean_geometry Euclidean geometry] any three [[points]], when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional [https://en.wikipedia.org/wiki/Euclidean_space Euclidean space]).

Triangles can be classified according to the relative lengths of their sides:

Triangles can be classified according to the relative lengths of their sides:

*In an [http://en.wikipedia.org/wiki/Equilateral_triangle equilateral triangle] all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.

*In an [https://en.wikipedia.org/wiki/Equilateral_triangle equilateral triangle] all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.

*In an '''isosceles triangle''', two sides are [[equal]] in length. An isosceles triangle also has two [[angles]] of the same measure; namely, the angles opposite to the two sides of the same length; this [[fact]] is the content of the [http://en.wikipedia.org/wiki/Isosceles_triangle_theorem isosceles triangle theorem], which was known by [http://en.wikipedia.org/wiki/Euclid Euclid]. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter [[definition]] would make all equilateral triangles isosceles triangles.  

*In an '''isosceles triangle''', two sides are [[equal]] in length. An isosceles triangle also has two [[angles]] of the same measure; namely, the angles opposite to the two sides of the same length; this [[fact]] is the content of the [https://en.wikipedia.org/wiki/Isosceles_triangle_theorem isosceles triangle theorem], which was known by [https://en.wikipedia.org/wiki/Euclid Euclid]. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter [[definition]] would make all equilateral triangles isosceles triangles.  

*In a scalene triangle, all sides are unequal, equivalently all angles are unequal. Right triangles are scalene if and only if not isosceles.

*In a scalene triangle, all sides are unequal, equivalently all angles are unequal. Right triangles are scalene if and only if not isosceles.

[[File:Tick_marks_triangle.jpg|center|frame]]

[[File:Tick_marks_triangle.jpg|center|frame]]

Triangles can also be classified according to their [http://en.wikipedia.org/wiki/Internal_angle internal angles], measured here in degrees.

Triangles can also be classified according to their [https://en.wikipedia.org/wiki/Internal_angle internal angles], measured here in degrees.

*A [http://en.wikipedia.org/wiki/Right_triangle right triangle] (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the [http://en.wikipedia.org/wiki/Hypotenuse hypotenuse]; it is the longest side of the right triangle. The other two sides are called the legs or cathet (singular: ''[http://en.wiktionary.org/wiki/cathetus cathetus]'') of the triangle. Right triangles obey the [http://en.wikipedia.org/wiki/Pythagorean_theorem Pythagorean theorem]: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where [[File:Pythtriple.jpg]]. In this situation, 3, 4, and 5 are a [http://en.wikipedia.org/wiki/Pythagorean_Triple Pythagorean Triple]. The other one is an isosceles triangle that has 2 angles that each measure 45 degrees.

*A [https://en.wikipedia.org/wiki/Right_triangle right triangle] (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the [https://en.wikipedia.org/wiki/Hypotenuse hypotenuse]; it is the longest side of the right triangle. The other two sides are called the legs or cathet (singular: ''[https://en.wiktionary.org/wiki/cathetus cathetus]'') of the triangle. Right triangles obey the [https://en.wikipedia.org/wiki/Pythagorean_theorem Pythagorean theorem]: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where [[File:Pythtriple.jpg]]. In this situation, 3, 4, and 5 are a [https://en.wikipedia.org/wiki/Pythagorean_Triple Pythagorean Triple]. The other one is an isosceles triangle that has 2 angles that each measure 45 degrees.

*Triangles that do not have an angle that measures 90° are called ''oblique triangles''.

*Triangles that do not have an angle that measures 90° are called ''oblique triangles''.

*A triangle that has all interior angles measuring less than 90° is an ''acute triangle'' or ''acute-angled triangle''.

*A triangle that has all interior angles measuring less than 90° is an ''acute triangle'' or ''acute-angled triangle''.

*A triangle that has one angle that measures more than 90° is an ''obtuse triangle'' or ''obtuse-angled triangle''.

*A triangle that has one angle that measures more than 90° is an ''obtuse triangle'' or ''obtuse-angled triangle''.

*A "triangle" with an interior angle of 180° (and [http://en.wiktionary.org/wiki/collinear collinear] vertices) is [http://en.wikipedia.org/wiki/Degeneracy_(mathematics) degenerate].

*A "triangle" with an interior angle of 180° (and [https://en.wiktionary.org/wiki/collinear collinear] vertices) is [https://en.wikipedia.org/wiki/Degeneracy_(mathematics) degenerate].

*A right degenerate triangle has collinear vertices, two of which are coincident.

*A right degenerate triangle has collinear vertices, two of which are coincident.

A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral.[http://en.wikipedia.org/wiki/Triangle]

A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral.[https://en.wikipedia.org/wiki/Triangle]

[[File:Oblique_triangle.jpg|center|frame]]

[[File:Oblique_triangle.jpg|center|frame]]