Difference between revisions of "Triangle"

Latest revision as of 18:06, 30 January 2025

A triangle is a of polygon with three sides.

$[asy]draw((0,1)--(2,0)--(3,2)--cycle);[/asy]$

A triangle

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.

Properties

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 $\frac{ab}{2} \sin C$ .
The area of any triangle with sides $a,b,c$ and semiperimeter $s$ is $\sqrt{s(s-a)(s-b)(s-c)}$ . This is known as Heron's Formula.
The area of any triangle with inradius $r$ and semiperimeter $s$ is $rs$ .
In any triangle, the sum of any two sides is greater than the length of the third side. This is known as the Triangle Inequality.
The sum of the interior angles of a triangle is $180^{\circ}$ .
See trigonometric identities for a list of formulae related to 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.
Mathematical Problems - Lecture delivered before the International Congress of Mathematicians at Paris in 1900 by David Hilbert.
Foundations of Geometry by David Hilbert

@@ Line 1: / Line 1: @@
-A '''triangle''' is a type of [[polygon]].
+A '''triangle''' is a of [[polygon]] with three [[edge|sides]].
-{{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle.}}
+{{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle}}
-==Definition==
-A '''triangle''' is any [[polygon]] with three [[edge | sides]], with the smaller angle measures of the intersections of the sides summing to 180 degrees. 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.
+== Definition ==
+A triangle is any [[polygon]] with three [[edge|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.
+== Categories ==
-==Categories==
 Triangles are split into six categories; three by their [[angle]]s and three by their side lengths.
-{{asy image|<asy>draw((0,0)--(1,0)--(0.5,0.5)--cycle);</asy>|right|An isosceles triangle.}}
 ===Equilateral===
 {{main|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 <math>60^{\circ}</math>
-===Isosceles===
+=== Isosceles ===
 {{main|Isosceles triangle}}
 An '''isosceles''' triangle has at least two congruent sides (this means that all equilateral triangles are also isosceles), and the two angles opposite the congruent sides are also congruent (this is commonly known as the [[Hinge theorem]]).
-===Scalene===
+=== Scalene ===
 {{main|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===
+=== Right ===
-{{asy image|<asy>draw((0,0)--(1,0)--(0,1)--cycle);</asy>|right|A right triangle.}}
 {{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.
+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.
-===Obtuse===
+=== Obtuse ===
+{{main|Obtuse triangle}}
 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===
+=== Acute ===
+{{main|Acute triangle}}
 All the angles of an '''acute''' triangle are [[acute angle]]s.
-==Related Formulas and Theorems==
+== Properties ==
-*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>
+* 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> is <math>\sqrt{s(s-a)(s-b)(s-c)}</math>, where <math>s</math> is the [[semiperimeter]] ([[Heron's Formula]]).
+* 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^2+b^2=c^2</math>. This is the famous [[Pythagorean theorem]].
+* The area of any triangle with sides <math>a,b,c</math> and semiperimeter <math>s</math> is <math>\sqrt{s(s-a)(s-b)(s-c)}</math>. This is known as [[Heron's Formula]].
-*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 area of any triangle with inradius <math>r</math> and semiperimeter <math>s</math> is <math>rs</math>.
-*In any triangle, the sum of any two sides is greater than the length of the third side.
+* In any triangle, the sum of any two sides is greater than the length of the third side. This is known as the [[Triangle Inequality]].
-*The sum of the interior angles of a triangle is <math>180^{\circ}</math>.
+* The sum of the interior angles of a triangle is <math>180^{\circ}</math>.
-*See [[trigonometric identities]] for a list of formulae related to trigonometry.
+* See [[trigonometric identities]] for a list of formulae related to trigonometry.
 == External Links ==
-* [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=9 Introduction to Geometry] by [[Richard Rusczyk]]
+* [{{SERVER}}/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.
@@ Line 46: / Line 57: @@
 * [http://www.gutenberg.org/files/17384/17384-pdf.pdf Foundations of Geometry] by David Hilbert
-==See Also==
+== See Also ==
 *[[Incircle]]
 *[[Excircle]]

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Difference between revisions of "Triangle"

Latest revision as of 18:06, 30 January 2025

Contents

Definition

Categories

Equilateral

Isosceles

Scalene

Right

Obtuse

Acute

Properties

External Links

See Also