Τρίγωνα

Triangle

From Wikipedia, the free encyclopedia

This article is about the basic geometric shape. For other uses, see Triangle (disambiguation).

"Isosceles" and "Acute Triangle" redirect here. For the trapezoid, see Isosceles trapezoid. For The Welcome to Paradox episode, see List of Welcome to Paradox episodes.

Triangle
A triangle
Edges and vertices	3
Schläfli symbol	{3} (for equilateral)
Area	various methods; see below
Internal angle (degrees)	60° (for equilateral)

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.
In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space).

Contents  [hide]  1 Types of triangles 1.1 By relative lengths of sides 1.2 By internal angles 2 Basic facts 3 Points, lines, and circles associated with a triangle 4 Computing the area of a triangle 4.1 Using vectors 4.2 Using trigonometry 4.3 Using coordinates 4.4 Using line integrals 4.5 Using Heron's formula 4.6 Formulas mimicking Heron's formula 4.7 Using Pick's Theorem 4.8 Other area formulas 4.9 Upper bound on the area 4.10 Bisecting the area 5 Computing the sides and angles 5.1 Trigonometric ratios in right triangles 5.1.1 Sine, cosine and tangent 5.1.2 Inverse functions 5.2 The sine, cosine and tangent rules 6 Further formulas for general Euclidean triangles 7 Morley's trisector theorem 8 Figures inscribed in a triangle 9 Non-planar triangles 10 Triangles in construction 11 See also 12 References 13 External links

[edit] Types of triangles

Euler diagram of types of triangles, using the definition that isosceles triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles.

[edit] By relative lengths of sides

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

• In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.^[1]
• In an isosceles triangle, two sides are equal in length.^[2][3] 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 Isosceles triangle theorem. 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.^[3] The latter definition would make all equilateral triangles isosceles triangles. The 45-45-90 Right Triangle, which appears in the Tetrakis square tiling, is isosceles.
• In a scalene triangle, all sides are unequal.^[4] The three angles are also all different in measure. Some (but not all) scalene triangles are also right triangles.

.

Equilateral	Isosceles	Scalene

In diagrams representing triangles (and other geometric figures), "tick" marks along the sides are used to denote sides of equal lengths-- the equilateral triangle has tick marks on all 3 sides, the isosceles on 2 sides. The scalene has single, double, and triple tick marks, indicating that no sides are equal. Similarly, arcs on the inside of the vertices are used to indicate equal angles. The equilateral triangle indicates all 3 angles are equal; the isosceles shows 2 identical angles. The scalene indicates by 1, 2, and 3 arcs that no angles are equal.

[edit] By internal angles

Triangles can also be classified according to their internal angles, measured here in degrees.

• A 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 hypotenuse; it is the longest side of the right triangle. The other two sides are called the legs or catheti^[5] (singular: cathetus) of the triangle. Right triangles obey the 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 3² + 4² = 5². In this situation, 3, 4, and 5 are a 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.

• 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" with an interior angle of 180° (and collinear vertices) is degenerate.

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.

Right	Obtuse	Acute
	Oblique