aarjaneiro/planetmath huggingface dataset:
compass and straightedge construction of regular pentagon

compass and straightedge construction of regular pentagon

One can construct a

regular

(

http://planetmath.org/RegularPolygon

)

pentagon

with sides of a given length

s

using

compass

and straightedge as follows:

1.

Draw a

line segment

of length

s

. Label its

endpoints

P

and

Q

.

2.

Extend the line segment past

Q

.

3.

Erect the

perpendicular

to

P

⁢

Q

→

at

Q

.

4.

Using the line drawn in the previous step, mark off a line segment of length

2

⁢

s

such that one of its endpoints is

Q

. Label the other endpoint as

R

.

5.

Connect

P

and

R

.

6.

Extend the line segment

P

⁢

R

¯

past

P

.

7.

On the extension, mark off another line segment of length

s

such that one of its endpoints is

P

. Label the other endpoint as

S

.

8.

Construct the

midpoint

of the line segment

R

⁢

S

¯

. Label it as

M

. (Below,

P

⁢

S

¯

is drawn in red, and

M

⁢

R

¯

is drawn in green.)

Note that the length of the line segment

M

⁢

R

¯

is

1

+

5

2

⁢

s

, which is the length of each

diagonal

of a

regular pentagon

with sides of length

s

.

9.

Separately from the drawing from the previous steps, draw a line segment of length

s

.

10.

Adjust the compass to the length of

M

⁢

R

¯

and draw an arc from each endpoint of the line segment from the previous step so that the arcs intersect.

11.

Adjust the compass to the length of

P

⁢

S

¯

and draw arcs from each of the three points to determine the other two points of the regular pentagon.

12.

Draw the regular pentagon.

The

law of cosines

can be used to justify this construction. Note that, in the picture below, the lengths of the line segments drawn in red are

s

and the lengths of the line segments drawn in green are

1

+

5

2

⁢

s

. The color of these line segments is based off of how the pentagon above was constructed.

If you are interested in seeing the rules for compass and straightedge constructions, click on the provided.

Title

compass and straightedge construction of regular pentagon

Canonical name

CompassAndStraightedgeConstructionOfRegularPentagon

Date of creation

2013-03-22 17:11:12

Last modified on

2013-03-22 17:11:12

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

27

Author

Wkbj79 (1863)

Entry type

Algorithm

Classification

msc 51-00

Classification

msc 51M15

Synonym

construction of regular pentagon

Related topic

RegularPolygon

Generated on Fri Feb 9 21:50:15 2018 by

LaTeXML
complete quadrilateral

complete quadrilateral

Let

A

⁢

B

⁢

C

⁢

D

be a

quadrilateral

. Let

{

F

}

=

A

⁢

B

⁢

⋂

C

⁢

D

and

{

E

}

=

B

⁢

C

⁢

⋂

A

⁢

D

. Then

A

⁢

F

⁢

C

⁢

E

is a

complete
quadrilateral

.

The complete quadrilateral has four sides :

A

⁢

B

⁢

F

¯

,

A

⁢

D

⁢

E

¯

,

B

⁢

C

⁢

E

¯

,

D

⁢

C

⁢

F

¯

, and six
angles:

A

,

B

,

C

,

D

,

E

,

F

.

Title

complete quadrilateral

Canonical name

CompleteQuadrilateral

Date of creation

2013-03-22 14:05:27

Last modified on

2013-03-22 14:05:27

Owner

vmoraru (1243)

Last modified by

vmoraru (1243)

Numerical id

5

Author

vmoraru (1243)

Entry type

Definition

Classification

msc 51-00

Generated on Fri Feb 9 21:50:17 2018 by

LaTeXML
concurrent

concurrent

A set of lines or curves is said to be

concurrent

if all of them

pass through

some point:

Title

concurrent

Canonical

name

Concurrent

Date of creation

2013-03-22 12:24:43

Last modified on

2013-03-22 12:24:43

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Synonym

concurrency

Related topic

PappussTheorem

Generated on Fri Feb 9 21:50:18 2018 by

LaTeXML
concyclic

concyclic

In any

geometry

where a circle is defined, a

collection

of points are said to be

concyclic

if there is a circle that is

incident

with all the points.

Remarks

.
Suppose all points being considered below lie in a

Euclidean plane

.

•

Any two points

P

,

Q

are concyclic. In fact, there are infinitely many circles that are incident to both

P

and

Q

. If

P

≠

Q

, then the pencil

𝔓

of circles incident with

P

and

Q

share the property that their centers are

collinear

. It is easy to see that any point on the

perpendicular bisector

of

P

⁢

Q

¯

serves as the center of a unique circle in

𝔓

.

•

Any three non-collinear points

P

,

Q

,

R

are concyclic to a unique circle

c

. From the three points, take any two perpendicular bisectors, say of

P

⁢

Q

¯

and

P

⁢

R

¯

. Then their

intersection

O

is the center of

c

, whose radius is

|

O

⁢

P

|

.

•

Four distinct points

A

,

B

,

C

,

D

are concyclic iff

∠

⁢

C

⁢

A

⁢

D

=

∠

⁢

C

⁢

B

⁢

D

.

Title

concyclic

Canonical name

Concyclic

Date of creation

2013-03-22 16:07:58

Last modified on

2013-03-22 16:07:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51-00

Generated on Fri Feb 9 21:50:27 2018 by

LaTeXML
converting between the Beltrami-Klein model and the Poincaré disc model

converting between the Beltrami-Klein model and the Poincaré disc model

The

Beltrami-Klein model

and the Poincaré disc model are closely related in that they both use the

unit disc

excluding the boundary and that poles can be defined for both models. In converting from one model to the other, it is easiest to use one unit disc for both models simultaneously. Lines of the two models can be put into correspondence as follows: Given a line

ℓ

in one model, its corresponding line

ℓ

′

in the other model is such that it has the same

endpoints

as

ℓ

.

If

ℓ

is a

diameter

, this means that

ℓ

′

=

ℓ

. Otherwise,

ℓ

′

will have the same pole as

ℓ

.

In all pictures in this entry, blue segments are lines in the Beltrami-Klein model, and red arcs are lines in the Poincaré disc model.

In the picture below, these two lines correspond to each other:

Given a line in the Poincaré disc model, it is easy to construct the corresponding line in the Beltrami-Klein model: Simply connect the endpoints by using a

straightedge

.

Given a line in the Beltrami-Klein model that is not a diameter of the

unit circle

, its pole must be found in to construct the corresponding line in the Poincaré disc model.

The pole serves as the center (

http://

planetmath

.org/Center8

) of the circle to which the corresponding line in the Poincaré disc model belongs.

Title

converting between the Beltrami-Klein model and the Poincaré disc model

Canonical name

ConvertingBetweenTheBeltramiKleinModelAndThePoincareDiscModel

Date of creation

2013-03-22 17:07:40

Last modified on

2013-03-22 17:07:40

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 51-00

Classification

msc 51M10

Related topic

BeltramiKleinModel2

Related topic

PoincareDiscModel

Generated on Fri Feb 9 21:50:34 2018 by

LaTeXML
crossed quadrilateral

crossed quadrilateral

A

complete crossed quadrilateral

is formed by four distinct lines

A

⁢

C

,

A

⁢

D

,

C

⁢

F

and

D

⁢

E

in the

Euclidean plane

, each of which

intersects

the other three. The

intersection

of

C

⁢

F

and

D

⁢

E

is labelled as

B

. A complete crossed quadrilateral has six vertices, of which

A

and

B

,

C

and

D

,

E

and

F

are opposite.

The complete crossed quadrilateral is often to the

crossed quadrilateral

C

⁢

E

⁢

D

⁢

F

(cyan in the diagram), consisting of the four

line segments

C

⁢

E

,

C

⁢

F

,

D

⁢

E

and

D

⁢

F

. Its

diagonals

C

⁢

D

and

E

⁢

F

are outside of the crossed quadrilateral. In the picture below, the same

quadrilateral

as above is still in cyan, and its diagonals are drawn in blue.

The sum of the inner angles of

C

⁢

E

⁢

D

⁢

F

is

720

o

. Its area is obtained e.g. (

http://planetmath.org/Eg

) by of the Bretschneider’s

formula

(cf. area of a quadrilateral).

A special case of the crossed quadrilateral is the

antiparallelogram

, in which the lengths of the opposite sides

C

⁢

E

and

D

⁢

F

are equal; similarly, the lengths of the opposite sides

C

⁢

F

and

D

⁢

E

are equal. Below, an antiparallelogram

C

⁢

E

⁢

D

⁢

F

is drawn in red. The antiparallelogram is with respect to the

perpendicular bisector

of the diagonal

C

⁢

D

(which is also the perpendicular bisector of the diagonal

E

⁢

F

). When the lengths of the sides

C

⁢

E

,

C

⁢

F

,

D

⁢

E

, and

D

⁢

F

are fixed, the

product

of the both diagonals

C

⁢

D

and

E

⁢

F

(yellow in the diagram) has a value, of the inner angles (e.g. on

α

).

Title

crossed quadrilateral

Canonical name

CrossedQuadrilateral

Date of creation

2013-03-22 17:11:34

Last modified on

2013-03-22 17:11:34

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

25

Author

pahio (2872)

Entry type

Definition

Classification

msc 51-00

Related topic

PtolemysTheorem

Defines

complete crossed quadrilateral

Defines

antiparallelogram

Generated on Fri Feb 9 21:50:36 2018 by

LaTeXML
cyclic quadrilateral

cyclic quadrilateral

Cyclic quadrilateral

.

A

quadrilateral

is cyclic when its four vertices lie on a circle.

A necessary and sufficient condition for a quadrilateral to be cyclic, is that the sum of a pair of opposite angles be equal to

180

∘

.

One of the main results about these quadrilaterals is Ptolemy’s theorem.

Also, from all the quadrilaterals with given sides

p

,

q

,

r

,

s

, the one that is cyclic has the greatest area. If the four sides of a cyclic quadrilateral are known, the area can be found using Brahmagupta’s formula

Title

cyclic quadrilateral

Canonical name

CyclicQuadrilateral

Date of creation

2013-03-22 11:44:16

Last modified on

2013-03-22 11:44:16

Owner

drini (3)

Last modified by

drini (3)

Numerical id

12

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Classification

msc 81R50

Classification

msc 81P05

Classification

msc 81Q05

Classification

msc 81-00

Synonym

cyclic

Related topic

OrthicTriangle

Related topic

PtolemysTheorem

Related topic

ProofOfPtolemysTheorem

Related topic

Circumcircle

Related topic

Quadrilateral

Generated on Fri Feb 9 21:50:38 2018 by

LaTeXML
derivation of cosines law

derivation of cosines law

The idea is to prove the

cosines law

:

a

2

=

b

2

+

c

2

-

2

⁢

b

⁢

c

⁢

cos

⁡

θ

where the

variables

are defined by the

triangle

:

{xy}

,

(

0

,

0

)

;

(

40

,

0

)

*

*

@

-

;

(

60

,

30

)

*

*

@

-

;

(

0

,

0

)

*

*

@

-

,

(

20

,

-

3

)

*

c

,

(

7

,

2

)

*

θ

,

(

50

,

12

)

*

a

,

(

30

,

17

)

*

b

Let’s add a couple of lines and two variables, to get

{xy}

,

(

0

,

0

)

;

(

40

,

0

)

*

*

@

-

;

(

60

,

30

)

*

*

@

-

;

(

0

,

0

)

*

*

@

-

,

(

20

,

-

3

)

*

c

,

(

7

,

2

)

*

θ

,

(

50

,

12

)

*

a

,

(

30

,

17

)

*

b

,

(

40

,

0

)

;

(

60

,

0

)

*

*

@

-

-

;

(

60

,

30

)

*

*

@

-

-

,

(

50

,

-

3

)

*

x

,

(

63

,

15

)

*

y

This is all we need. We can use Pythagoras’

theorem

to show that

a

2

=

x

2

+

y

2

and

b

2

=

y

2

+

(

c

+

x

)

2

So, combining these two we get

a

2

=

x

2

+

b

2

-

(

c

+

x

)

2

a

2

=

x

2

+

b

2

-

c

2

-

2

⁢

c

⁢

x

-

x

2

a

2

=

b

2

-

c

2

-

2

⁢

c

⁢

x

So, all we need now is an

expression

for

x

. Well, we can use the

definition

of the cosine function to show that

c

+

x

=

b

⁢

cos

⁡

θ

x

=

b

⁢

cos

⁡

θ

-

c

With this result in hand, we find that

a

2

=

b

2

-

c

2

-

2

⁢

c

⁢

x

a

2

=

b

2

-

c

2

-

2

⁢

c

⁢

(

b

⁢

cos

⁡

θ

-

c

)

a

2

=

b

2

-

c

2

-

2

⁢

b

⁢

c

⁢

cos

⁡

θ

+

2

⁢

c

2

a

2

=

b

2

+

c

2

-

2

⁢

b

⁢

c

⁢

cos

⁡

θ

(1)

Title

derivation of cosines law

Canonical

name

DerivationOfCosinesLaw

Date of creation

2013-03-22 11:57:02

Last modified on

2013-03-22 11:57:02

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Proof

Classification

msc 51-00

Related topic

CosinesLaw

Related topic

ProofOfCosinesLaw

Generated on Fri Feb 9 21:50:40 2018 by

LaTeXML
determining when a point is inside a polygon

determining when a point is inside a polygon

There are many

algorithms

to determine whether a point is inside a

polygon

or not:

•

Connect the

point at infinity

with a line. If this line
crosses the edges of the polygon an

odd number

of times, the point
is inside the polygon. Otherwise it is not.

•

Calculate

the winding number of the polygon with the
point.

These two algorithms can be used for any

closed curve

.

Title

determining when a point is inside a polygon

Canonical name

DeterminingWhenAPointIsInsideAPolygon

Date of creation

2013-03-22 15:22:24

Last modified on

2013-03-22 15:22:24

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Topic

Classification

msc 51-00

Related topic

WhenIsAPointInsideATriangle

Generated on Fri Feb 9 21:50:42 2018 by

LaTeXML
diameter

diameter

The

diameter

of a circle or a sphere is the length of the segment joining a point with the one symmetrical respect to the center. That is, the length of the longest segment joining a pair of points.

Also, we call any of these segments themselves a diameter. So, in the next picture,

A

⁢

B

is a diameter.

The diameter is equal to twice the radius.

Title

diameter

Canonical

name

Diameter1

Date of creation

2013-03-22 12:20:42

Last modified on

2013-03-22 12:20:42

Owner

drini (3)

Last modified by

drini (3)

Numerical id

20

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Related topic

Radius

Related topic

VolumeOfSphericalCapAndSphericalSector

Generated on Fri Feb 9 21:50:44 2018 by

LaTeXML
endpoint

endpoint

An

endpoint

of a

line segment

L

is a point that belongs to the boundary of

L

. Note that every line segment has two distinct endpoints. For example, if

V

is a vector space and

a

,

b

∈

V

with

b

≠

0

, then the endpoints of the line segment

L

=

{

a

+

t

⁢

b

:

t

∈

[

0

,

1

]

}

are

a

and

a

+

b

.

Note that the endpoints of the

open line segment

L

=

{

a

+

t

⁢

b

:

t

∈

(

0

,

1

)

}

are also

a

and

a

+

b

.

Endpoints can be defined in a manner for other geometric . These include rays, arcs (

http://planetmath.org/PathConnected

), and

intervals

(

http://planetmath.org/Interval

).

•

Rays have one endpoint.

•

have two endpoints.

•

can have zero, one, or two endpoints, depending on whether they are bounded above and/or below. See the entry on intervals (

http://planetmath.org/Interval

) for more details.

Title

endpoint

Canonical name

Endpoint

Date of creation

2013-03-22 16:06:05

Last modified on

2013-03-22 16:06:05

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

12

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Classification

msc 03-00

Synonym

end point

Generated on Fri Feb 9 21:50:50 2018 by

LaTeXML
equiangular polygon

equiangular polygon

A

polygon

is

equiangular

if all of its

interior angles

are

congruent

.

Common examples of

equiangular polygons

are

rectangles

and

regular polygons

such as

equilateral triangles

and squares.

Let

T

be a

triangle

in

Euclidean geometry

,

hyperbolic geometry

, or

spherical geometry

. Then the following are

equivalent

:

•

T

is

equilateral

;

•

T

is equiangular;

•

T

is

regular

.

If

T

is allowed to be a polygon that has more than three sides, then the above statement is no longer true in any of the indicated

geometries

.

Below are some pictures of equiangular polygons drawn in Euclidean geometry that are not equilateral.

Title

equiangular polygon

Canonical

name

EquiangularPolygon

Date of creation

2013-03-22 17:12:43

Last modified on

2013-03-22 17:12:43

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Synonym

equiangular

Related topic

BasicPolygon

Generated on Fri Feb 9 21:50:53 2018 by

LaTeXML
equiangular triangle

equiangular triangle

An

equiangular triangle

is one for which all three

interior angles

are

congruent

.

By the

theorem

at determining from angles that a

triangle

is

isosceles

, we can conclude that, in any

geometry

in which ASA holds, an

equilateral triangle

is

regular

(

http://planetmath.org/RegularTriangle

). In any geometry in which ASA, SAS, SSS, and AAS all hold, the

isosceles triangle theorem

yields that the

bisector

of any angle of an equiangular triangle coincides with the height, the median and the

perpendicular bisector

of the opposite side.

The following statements hold in

Euclidean geometry

for an equiangular triangle.

•

The triangle is determined by specifying one side.

•

If

r

is the length of the side, then the height is equal to

r

⁢

3

2

.

•

If

r

is the length of the side, then the area is equal to

r

2

⁢

3

4

.

Title

equiangular triangle

Canonical

name

EquiangularTriangle

Date of creation

2013-03-22 17:12:50

Last modified on

2013-03-22 17:12:50

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Related topic

Triangle

Related topic

IsoscelesTriangle

Related topic

EquilateralTriangle

Related topic

RegularTriangle

Related topic

EquivalentConditionsForTriangles

Generated on Fri Feb 9 21:50:56 2018 by

LaTeXML
equilateral polygon

equilateral polygon

A

polygon

is

equilateral

if all of its sides are

congruent

.

Common examples of

equilateral polygons

are rhombi and

regular polygons

such as

equilateral triangles

and squares.

Let

T

be a

triangle

in

Euclidean geometry

,

hyperbolic geometry

, or

spherical geometry

. Then the following are

equivalent

:

•

T

is equilateral;

•

T

is

equiangular

;

•

T

is

regular

.

If

T

is allowed to be a polygon that has more than three sides, then the above statement is no longer true in any of the indicated

geometries

.

Below are some pictures of equilateral polygons that are not equiangular.

Title

equilateral polygon

Canonical

name

EquilateralPolygon

Date of creation

2013-03-22 17:12:41

Last modified on

2013-03-22 17:12:41

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Synonym

equilateral

Related topic

BasicPolygon

Generated on Fri Feb 9 21:50:58 2018 by

LaTeXML
equilateral triangle

equilateral triangle

An

equilateral triangle

is one for which all 3 sides are

congruent

.

The following statements hold in

Euclidean geometry

for an equilateral triangle.

•

It is a

regular polygon

.

•

The

bisector

of any angle coincides with the height, the median and the

perpendicular bisector

of the .

•

If

r

is the length of the side, then the height is equal to

r

⁢

3

2

.

•

If

r

is the length of the side, then the area is equal to

r

2

⁢

3

4

.

Title

equilateral triangle

Canonical name

EquilateralTriangle

Date of creation

2013-03-22 11:44:04

Last modified on

2013-03-22 11:44:04

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

22

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 51-00

Classification

msc 81-00

Classification

msc 08-01

Related topic

Triangle

Related topic

IsoscelesTriangle

Related topic

EquivalentConditionsForTriangles

Related topic

EquiangularTriangle

Related topic

RegularTriangle

Generated on Fri Feb 9 21:51:01 2018 by

LaTeXML
equivalent conditions for triangles

equivalent conditions for triangles

The following

theorem

holds in

Euclidean geometry

,

hyperbolic geometry

, and

spherical geometry

:

Theorem 1

.

Let

△

⁢

A

⁢

B

⁢

C

be a

triangle

. Then the following are

equivalent

:

•

△

⁢

A

⁢

B

⁢

C

is

equilateral

(

http://planetmath.org/EquilateralTriangle

);

•

△

⁢

A

⁢

B

⁢

C

is

equiangular

(

http://planetmath.org/EquiangularTriangle

);

•

△

⁢

A

⁢

B

⁢

C

is

regular

(

http://planetmath.org/RegularTriangle

).

Note that this statement does not generalize to any

polygon

with more than three sides in any of the indicated

geometries

.

Proof.

It suffices to show that

△

⁢

A

⁢

B

⁢

C

is equilateral if and only if it is equiangular.

Sufficiency

: Assume that

△

⁢

A

⁢

B

⁢

C

is equilateral.

Since

A

⁢

B

¯

≅

A

⁢

C

¯

≅

B

⁢

C

¯

, SSS yields that

△

⁢

A

⁢

B

⁢

C

≅

△

⁢

B

⁢

C

⁢

A

. By CPCTC,

∠

⁢

A

≅

∠

⁢

B

≅

∠

⁢

C

. Hence,

△

⁢

A

⁢

B

⁢

C

is equiangular.

Necessity: Assume that

△

⁢

A

⁢

B

⁢

C

is equiangular.

By the theorem on determining from angles that a triangle is

isosceles

, we conclude that

△

⁢

A

⁢

B

⁢

C

is isosceles with legs

A

⁢

B

¯

≅

A

⁢

C

¯

and that

△

⁢

B

⁢

C

⁢

A

is isosceles with legs

A

⁢

C

¯

≅

B

⁢

C

¯

. Thus,

A

⁢

B

¯

≅

A

⁢

C

¯

≅

B

⁢

C

¯

. Hence,

△

⁢

A

⁢

B

⁢

C

is equilateral.
∎

Title

equivalent conditions for triangles

Canonical name

EquivalentConditionsForTriangles

Date of creation

2013-03-22 17:12:46

Last modified on

2013-03-22 17:12:46

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 51-00

Related topic

Triangle

Related topic

IsoscelesTriangle

Related topic

EquilateralTriangle

Related topic

EquiangularTriangle

Related topic

RegularTriangle

Generated on Fri Feb 9 21:51:03 2018 by

LaTeXML
Erdős-Anning Theorem

Erdős-Anning Theorem

If an infinite number of points in a plane are all separated by integer

distances

, then all the points lie on a

straight line

.

Title

Erdős-Anning Theorem

Canonical name

ErdHosAnningTheorem

Date of creation

2013-03-22 13:18:31

Last modified on

2013-03-22 13:18:31

Owner

giri (919)

Last modified by

giri (919)

Numerical id

5

Author

giri (919)

Entry type

Theorem

Classification

msc 51-00

Generated on Fri Feb 9 21:51:05 2018 by

LaTeXML
Euler line

Euler line

In any

triangle

, the

orthocenter

H

, the

centroid

G

and the

circumcenter

O

are

collinear

, and

O

⁢

G

/

G

⁢

H

=

1

/

2

. The line passing by these points is known as the

Euler line

of the triangle.

This line also passes by the center of the

nine-point circle

(or

Feuerbach circle

)

N

, and

N

is the

midpoint

of

O

⁢

H

.

Title

Euler line

Canonical name

EulerLine

Date of creation

2013-03-22 11:44:25

Last modified on

2013-03-22 11:44:25

Owner

drini (3)

Last modified by

drini (3)

Numerical id

18

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Classification

msc 58A05

Classification

msc 83E15

Classification

msc 83D05

Classification

msc 83E05

Classification

msc 83E50

Classification

msc 81-00

Classification

msc 83-00

Classification

msc 82-00

Related topic

Triangle

Related topic

Orthocenter

Related topic

Centroid

Related topic

Collinear

Related topic

Midpoint

Related topic

OrthicTriangle

Related topic

CenterOfATriangle

Related topic

EulerLineProof

Generated on Fri Feb 9 21:51:07 2018 by

LaTeXML