triangle

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

∘

. In the figure:

A

+

B

+

C

=

180

∘

.

In

hyperbolic geometry

, the angle sum of a triangle is always

strictly

positive

and strictly less than

180

∘

. In the figure:

0

∘

<

A

+

B

+

C

<

180

∘

.

In

spherical geometry

, the angle sum of a triangle is always strictly

greater

than

180

∘

and strictly less than

540

∘

. In the figure:

180

∘

<

A

+

B

+

C

<

540

∘

.

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

∘

.

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

A

=

1

2

⁢

b

⁢

h

, 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

s

=

a

+

b

+

c

2

be the

semiperimeter

. Then

Area

=

a

⁢

h

a

2

=

b

⁢

h

b

2

=

c

⁢

h

c

2

=

a

⁢

b

⁢

sin

⁡

C

2

=

b

⁢

c

⁢

sin

⁡

A

2

=

c

⁢

a

⁢

sin

⁡

B

2

=

a

⁢

b

⁢

c

4

⁢

R

=

s

⁢

r

=

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

±

1

2

⁢

|

x

1

y

1

1

x

2

y

2

1

x

3

y

3

1

|

(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

π

radians (

180

∘

)

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

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