Difference between revisions of "Triangle"

Revision as of 21:20, 9 November 2007

A triangle is a type of polygon.

Definition

A triangle is any polygon with three sides. Triangles exist in Euclidean geometry, and are the simplest possible polygon. In physics, triangles are noted for their durability, since they have only three vertices around with to distort.

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Categories

Triangles are split into six categories; three by their angles and three by their side lengths.

Equilateral

Main article: Equilateral triangle

An equilateral triangle has three congruent sides, and is also equiangular. Note that all equilateral triangles are similar. All the angles of equilateral triangles are $60^{\circ}$

Isoceles

Main article: Isoceles triangle

An isoceles triangle has at least two congruent sides (this means that all equilateral triangles are also isoceles), and the two angles opposite the congruent sides are also congruent (this is commonly known as the base angles theorem).

Scalene

Main article: Scalene triangle

A scalene triangle has no congruent sides, and also no congruent angles (this can be proven by the converse of the Hinge Theorem.

Right

Main article: Right triangle

A right triangle has a right angle, which means the other two angles are complementary. Trigonometry is largely based on right triangles, and the famous Pythagorean Theorem deals with the side lengths of the right triangle. Note that it is impossible to have an equilateral right triangle. It is also impossible to have more than one right angle in a triangle.

Obtuse

An obtuse triangle has an obtuse angle. Note that it is impossible to have an equilateral obtuse triangle. It is also impossible to have more than one obtuse angle in a triangle.

Acute

All the angles of an acute triangle are acute angles.

Related Formulae and Theorems

The area of any triangle with base $b$ and height $h$ is $\frac{bh}{2}$ . (This can be shown by combining the triangle and a copy of it into a parallelogram).
The area of any triangle with sides $a,b,c$ opposite angles $A,B,C$ is

$\frac {ab}{2}\sin C$

For a right triangle with legs $a,b$ and hypotenuse $c$ , $a^+b^2=c^2$ . This is the famous Pythagorean theorem.
The inradius of a triangle with sides $a,b,c$ and area $K$ is $\frac{2K}{a+b+c}$ .
The sum of the interior angles of a triangle is $180^{\circ}$ .
See trigonometric identities for a list of formulae related the trigonometry.

External Links

Introduction to Geometry by Richard Rusczyk
Challenging Problems in Geometry - A good book for students who already have a solid handle on elementary geometry.
Geometry Revisited - A classic.

@@ Line 1: / Line 1: @@
-A '''triangle''' is a [[polygon]] with three sides.
+A '''triangle''' is a type of [[polygon]].
+==Definition==
+A triangle is any polygon with three [[side]]s. Triangles exist in Euclidean [[geometry]], and are the simplest possible polygon. In [[physics]], triangles are noted for their durability, since they have only three [[vertex|vertices]] around with to distort.
+{{image}}
-== Introductory topics ==
+==Categories==
-* [[Congruence]]
+Triangles are split into six categories; three by their [[angle]]s and three by their side lengths.
-* [[Similarity]]
+===Equilateral===
-* [[Triangle inequality]]
+{{main|Equilateral triangle}}
-* [[Area]]
+An '''equilateral''' triangle has three congruent sides, and is also [[equiangular]]. Note that all equilateral triangles are [[similar]]. All the angles of equilateral triangles are <math>60^{\circ}</math>
-* [[Angle bisector]]s
+===Isoceles===
-* [[Incircle]]
+{{main|Isoceles triangle}}
-* [[Circumcircle]]
+An '''isoceles''' triangle has at least two congruent sides (this means that all equilateral triangles are also isoceles), and the two angles opposite the congruent sides are also congruent (this is commonly known as the base angles theorem).
-* [[Centroid]]
+===Scalene===
-=== Resources ===
+{{main|Scalene triangle}}
-==== Books ====
+A '''scalene''' triangle has no congruent sides, and also no congruent angles (this can be proven by the converse of the [[Hinge Theorem]].
+===Right===
+{{main|Right triangle}}
+A '''right''' triangle has a [[right angle]], which means the other two angles are [[complementary]]. [[Trigonometry]] is largely based on right triangles, and the famous [[Pythagorean Theorem]] deals with the side lengths of the right triangle. Note that it is impossible to have an equilateral right triangle. It is also impossible to have more than one right angle in a triangle.
+===Obtuse===
+An '''obtuse''' triangle has an [[obtuse angle]]. Note that it is impossible to have an equilateral obtuse triangle. It is also impossible to have more than one obtuse angle in a triangle.
+===Acute===
+All the angles of an '''acute''' triangle are [[acute angle]]s.
+==Related Formulae and Theorems==
+*The [[area]] of any triangle with base <math>b</math> and height <math>h</math> is <math>\frac{bh}{2}</math>. (This can be shown by combining the triangle and a copy of it into a [[parallelogram]]).
+*The [[area]] of any triangle with sides <math>a,b,c</math> opposite angles <math>A,B,C</math> is
+<math>\frac {ab}{2}\sin C </math>
+*For a right triangle with legs <math>a,b</math> and [[hypotenuse]] <math>c</math>, <math>a^+b^2=c^2</math>. This is the famous [[Pythagorean theorem]].
+*The [[inradius]] of a triangle with sides <math>a,b,c</math> and area <math>K</math> is <math>\frac{2K}{a+b+c}</math>.
+*The sum of the interior angles of a triangle is <math>180^{\circ}</math>.
+*See [[trigonometric identities]] for a list of formulae related the trigonometry.
+== External Links ==
 * [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=9 Introduction to Geometry] by [[Richard Rusczyk]]
+* [http://www.amazon.com/exec/obidos/ASIN/0486691543/artofproblems-20 Challenging Problems in Geometry] - A good book for students who already have a solid handle on elementary geometry.
+* [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] - A classic.
+==See Also==
+*[[Incircle]]
+*[[Excircle]]
+*[[Circumcircle]]
+*[[Similarity]]
+*[[Congruence]]
-== Intermediate topics ==
+[[Category:Definition]]
-* [[Geometric inequalities]]
-=== Resources ===
-==== Books ====
-* [http://www.amazon.com/exec/obidos/ASIN/0486691543/artofproblems-20 Challenging Problems in Geometry] -- A good book for students who already have a solid handle on elementary geometry.
-* [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] -- A classic.
 [[Category:Geometry]]
+[[Category:Polygon]]

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