aarjaneiro/planetmath huggingface dataset:
median

median

The

median

of a

triangle

is a

line segment

joining a vertex with the

midpoint

of the opposite side.

In the next figure,

A

⁢

A

′

is a median. That is,

B

⁢

A

′

=

A

′

⁢

C

, or equivalently,

A

′

is the midpoint of

B

⁢

C

.

If the length of the three sides of the triangle are known, the length of the medians can be found by means of

Apollonius theorem

.

Title

median

Canonical name

Median

Date of creation

2013-03-22 11:44:01

Last modified on

2013-03-22 11:44:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

18

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51-00

Classification

msc 55-00

Classification

msc 55-01

Related topic

Triangle

Related topic

ApolloniusTheorem

Related topic

Orthocenter

Related topic

CevasTheorem

Related topic

Centroid

Related topic

ProofOfApolloniusTheorem2

Related topic

ParallelogramLaw

Related topic

TrigonometricVersionOfCevasTheorem

Related topic

ProofOfParallelogramLaw

Related topic

HeightOfATriangle

Related topic

Cevian

Generated on Fri Feb 9 21:51:57 2018 by

LaTeXML
Miquel point

Miquel point

Let

A

⁢

E

⁢

C

⁢

F

be a

complete quadrilateral

, then the four circles
circumcised to the four

triangles

:

A

⁢

E

⁢

D

,

A

⁢

F

⁢

B

,

B

⁢

E

⁢

C

,

C

⁢

D

⁢

F

,
are

concurrent

in a point

M

. This point is called the

Miquel point

.

The Miquel point is also the focus of the

parabola

inscribed

to

A

⁢

E

⁢

C

⁢

F

.

Title

Miquel point

Canonical name

MiquelPoint

Date of creation

2013-03-22 14:05:30

Last modified on

2013-03-22 14:05:30

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 21:52:06 2018 by

LaTeXML
Mollweide’s equations

Mollweide’s equations

[10pt]

In a triangle, having the sides

a

,

b

and

c

opposite to the angles

α

,

β

and

γ

respectively the following equations hold:

(

a

+

b

)

⁢

sin

⁡

γ

2

=

c

⁢

cos

⁡

(

α

-

β

2

)

and

(

a

-

b

)

⁢

cos

⁡

γ

2

=

c

⁢

sin

⁡

(

α

-

β

2

)

.

Title

Mollweide’s equations

Canonical name

MollweidesEquations

Date of creation

2013-03-22 12:50:08

Last modified on

2013-03-22 12:50:08

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

5

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 51-00

Generated on Fri Feb 9 21:52:08 2018 by

LaTeXML
Morley’s theorem

Morley’s theorem

Theorem 1

(Morley’s Theorem)

The points of intersections of the

adjacent

trisectors in any

triangle

, are the vertices of an

equilateral triangle

.

Title

Morley’s theorem

Canonical name

MorleysTheorem

Date of creation

2013-03-22 11:43:58

Last modified on

2013-03-22 11:43:58

Owner

drini (3)

Last modified by

drini (3)

Numerical id

13

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Classification

msc 55U10

Classification

msc 55P20

Classification

msc 55-00

Classification

msc 55U15

Classification

msc 55U99

Classification

msc 55U40

Related topic

Triangle

Generated on Fri Feb 9 21:52:10 2018 by

LaTeXML
Newton-Gauss line

Newton-Gauss line

Let

A

⁢

E

⁢

C

⁢

F

be a

complete quadrilateral

, and

A

⁢

C

,

B

⁢

D

,

E

⁢

F

his

diagonals

. Let

P

be the middle of

A

⁢

C

,

Q

the middle of

B

⁢

D

,
and

R

the middle of

E

⁢

F

. Then

P

,

Q

,

R

are on a same line,
called the

Newton-Gauss line

.

Title

Newton-Gauss line

Canonical name

NewtonGaussLine

Date of creation

2013-03-22 14:05:47

Last modified on

2013-03-22 14:05:47

Owner

vmoraru (1243)

Last modified by

vmoraru (1243)

Numerical id

4

Author

vmoraru (1243)

Entry type

Definition

Classification

msc 51-00

Generated on Fri Feb 9 21:52:11 2018 by

LaTeXML
Newton’s line

Newton’s line

Let

A

⁢

B

⁢

C

⁢

D

be a circumscribed quadrilateral. The middle of the two

diagonals

M

,

N

and the center of the inscribed circle

I

are

collinear

. This line is called the

Newton’s line

Title

Newton’s line

Canonical name

NewtonsLine

Date of creation

2013-03-22 14:05:34

Last modified on

2013-03-22 14:05:34

Owner

vmoraru (1243)

Last modified by

vmoraru (1243)

Numerical id

4

Author

vmoraru (1243)

Entry type

Definition

Classification

msc 51-00

Generated on Fri Feb 9 21:52:13 2018 by

LaTeXML
nine-point circle

nine-point circle

The

nine

point circle

also known as the

Euler’s circle

or the

Feuerbach circle

is the circle that

passes through

the feet of

perpendiculars

from the vertices

A

,

B

and

C

of a

triangle

△

⁢

A

⁢

B

⁢

C

.

Some of the

properties

of this circle are:

Property 1 :

This circle also passes through the

midpoints

of the sides

A

⁢

B

,

B

⁢

C

and

C

⁢

A

of

△

⁢

A

⁢

B

⁢

C

.

This was shown by Euler.

Property 2 :

Feuerbach showed that this circle also passes through the midpoints of the

line segments

A

⁢

H

,

B

⁢

H

and

C

⁢

H

which are drawn from the vertices of

△

⁢

A

⁢

B

⁢

C

to its

orthocenter

H

.

These three triples of points make nine in all, giving the circle its name.

Property 3 :

The radius of the nine-point cirlce is

R

/

2

,

where

R

is the

circumradius

(radius of the

circumcircle

).

Property 4 :

The center of the

nine-point circle

is the midpoint of the line segment joining the orthocenter and the

circumcenter

, and hence lies on the

Euler line

.

Property 5 :

All triangles

inscribed

in a given circle and having the same orthocenter, have the same nine-point circle.

Title

nine-point circle

Canonical

name

NinepointCircle

Date of creation

2013-03-22 13:11:20

Last modified on

2013-03-22 13:11:20

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Definition

Classification

msc 51-00

Synonym

Euler circle

Synonym

Feuerbach circle

Synonym

nine point circle

Generated on Fri Feb 9 21:52:14 2018 by

LaTeXML
opposite

opposite

Given a

right triangle

with an

acute angle

θ

, the side of the triangle that is

opposite

of

θ

is the side of the triangle that is not a side (

http://planetmath.org/Angle

) of

θ

.

This

definition

can be generalized to any triangle and any angle

θ

: The side of the triangle that is

opposite

of

θ

is the side of the triangle that is not a side of

θ

. This more general definition is not used as commonly as the definition for right triangles.

Title

opposite

Canonical

name

Opposite

Date of creation

2013-03-22 15:59:28

Last modified on

2013-03-22 15:59:28

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Classification

msc 51-01

Synonym

opposite side

Related topic

Sohcahtoa

Generated on Fri Feb 9 21:52:18 2018 by

LaTeXML
origin

origin

In the

vector space

ℝ

n

, the word

origin

refers to the
zero point, that is the point

(

0

,

…

,

0

)

. Similarly for

ℂ

n

.

definitions

can be made for any vector space. Often the notation

0

or

O

is used for the origin.

In some contexts the choice of “origin” can be arbitrary and thus not natural.
For example, if we think of

Euclidean space

as an affine space or as a

Riemannian manifold

, it has no natural origin.
Many

theorems

about local properties of

manifolds

are stated for values near the origin in some vector space. This is because any point on the manifold can be the origin in some set of local coordinates.

Title

origin

Canonical

name

Origin

Date of creation

2013-03-22 15:04:31

Last modified on

2013-03-22 15:04:31

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 51-00

Generated on Fri Feb 9 21:52:19 2018 by

LaTeXML
orthic triangle

orthic triangle

If

A

⁢

B

⁢

C

is a

triangle

and

A

⁢

D

,

D

⁢

E

,

C

⁢

F

are its three heights (

http://planetmath.org/BaseAndHeightOfTriangle

), then the triangle

D

⁢

E

⁢

F

is called the

orthic triangle

of

A

⁢

B

⁢

C

.

A remarkable

property

of orthic triangles says that the

orthocenter

of

A

⁢

B

⁢

C

is also the

incenter

of the orthic triangle

D

⁢

E

⁢

F

. That is, the heights of

A

⁢

B

⁢

C

are the

angle bisectors

of

D

⁢

E

⁢

F

.

Title

orthic triangle

Canonical

name

OrthicTriangle

Date of creation

2013-03-22 12:11:00

Last modified on

2013-03-22 12:11:00

Owner

drini (3)

Last modified by

drini (3)

Numerical id

8

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Related topic

Triangle

Related topic

Orthocenter

Related topic

EulerLine

Related topic

CevasTheorem

Related topic

CyclicQuadrilateral

Related topic

TrigonometricVersionOfCevasTheorem

Related topic

BaseAndHeightOfTriangle

Generated on Fri Feb 9 21:52:21 2018 by

LaTeXML
orthocenter

orthocenter

The

orthocenter

of a

triangle

is the point of

intersection

of its three heights.

In the figure,

H

is the orthocenter of

A

⁢

B

⁢

C

.

The orthocenter

H

lies inside, on a vertex or outside the triangle depending on the triangle being acute, right or obtuse respectively. Orthocenter is one of the most important

triangle centers

and it is very related with the

orthic triangle

(formed by the three height’s foots). It lies on the

Euler line

and the four

quadrilaterals

F

⁢

H

⁢

D

⁢

B

,

C

⁢

H

⁢

E

⁢

C

,

A

⁢

F

⁢

H

⁢

E

are cyclic.

In fact,

•

A

is the orthocenter of

B

,

C

,

H

;

•

B

is the orthocenter of

A

,

C

,

H

;

•

C

is the orthocenter of

A

,

B

,

H

.

The four points

A

,

B

,

C

, and

H

form what is called
an

orthocentric tetrad

.

Title

orthocenter

Canonical name

Orthocenter

Date of creation

2013-03-22 11:55:41

Last modified on

2013-03-22 11:55:41

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 51-00

Related topic

HeightOfATriangle

Related topic

Median

Related topic

Triangle

Related topic

EulerLine

Related topic

OrthicTriangle

Related topic

CEvasTheorem

Related topic

CevasTheorem

Related topic

CenterOfATriangle

Related topic

Incenter

Related topic

TrigonometricVersionOfCevasTheorem

Related topic

Centroid

Defines

orthocentric tetrad

Generated on Fri Feb 9 21:52:23 2018 by

LaTeXML
parallel lines in hyperbolic geometry

parallel lines in hyperbolic geometry

In

hyperbolic geometry

, there are two kinds of

parallel lines

. If two lines do not intersect within a model of hyperbolic geometry but they do intersect on its boundary, then the lines are called

asymptotically parallel

or

hyperparallel

. (Note that, in the

upper half plane model

, any two vertical rays are asymptotically parallel. Thus, for consistency,

∞

is considered to be part of the boundary.) Any other set of parallel lines is called

disjointly parallel

or

ultraparallel

.

Below is an example of asymptotically parallel lines in the

Beltrami-Klein model

:

Below are some examples of asymptotically parallel lines in the Poincaré disc model:

Below are some examples of asymptotically parallel lines in the upper half plane model:

Below is an example of disjointly parallel lines in the Beltrami-Klein model:

Below is an example of disjointly parallel lines in the Poincaré disc model:

Below are some examples of disjointly parallel lines in the upper half plane model:

Title

parallel lines in hyperbolic geometry

Canonical name

ParallelLinesInHyperbolicGeometry

Date of creation

2013-03-22 17:06:43

Last modified on

2013-03-22 17:06:43

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 51-00

Classification

msc 51M10

Defines

asymptotically parallel

Defines

asymptotically parallel lines

Defines

hyperparallel

Defines

hyperparallel lines

Defines

disjointly parallel

Defines

disjointly parallel lines

Defines

ultraparallel

Defines

ultraparallel lines

Generated on Fri Feb 9 21:52:25 2018 by

LaTeXML
parallelogram

parallelogram

A

parallelogram

is a

quadrilateral

whose opposite sides are

parallel

.

Some special parallelograms have their own names: squares,

rectangles

,

rhombuses

.
A rectangle is a parallelogram whose all angles are equal (i.e.

90

∘

), a

rhombus

is a parallelogram whose all sides are equal, and a square is a parallelogram that is a rectangle and a rhombus at the same time.

Every parallelogram have their opposite sides and opposite angles equal. Also,

adjacent

angles of a parallelogram always add up to

180

∘

, and the

diagonals

bisect each other.

A

base

of a parallelogram is one of its sides. Any side of a
parallelogram can be selected as its base. Once a base has been
selected, the

height

of the parallelogram is defined to be the
length of any

line segment

perpendicular

to the base which extends
from the base to the opposite side parallel to the base. The area of
a parallelogram with base length

b

and height

h

is given by the
formula

A

=

b

⁢

h

.

There is also a neat relation between the length of the sides and the lengths of the diagonals called the

parallelogram law

.

Title

parallelogram

Canonical name

Parallelogram

Date of creation

2013-03-22 12:02:21

Last modified on

2013-03-22 12:02:21

Owner

drini (3)

Last modified by

drini (3)

Numerical id

11

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Related topic

ParallelogramTheorems

Related topic

Quadrilateral

Related topic

Rectangle

Related topic

Rhombus

Related topic

Square

Related topic

ParallelogramLaw

Related topic

Kite

Related topic

Rhomboid

Related topic

ProofOfParallelogramTheorems

Related topic

Parallelotope

Generated on Fri Feb 9 21:52:27 2018 by

LaTeXML
parallelogram law

parallelogram law

Let

A

⁢

B

⁢

C

⁢

D

be a

parallelogram

with side lengths

u

,

v

and whose

diagonals

have lengths

d

1

and

d

2

then

2

⁢

u

2

+

2

⁢

v

2

=

d

1

2

+

d

2

2

.

Title

parallelogram law

Canonical name

ParallelogramLaw

Date of creation

2013-03-22 12:02:23

Last modified on

2013-03-22 12:02:23

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Related topic

Parallelogram

Related topic

Quadrilateral

Related topic

Rectangle

Related topic

Square

Related topic

Rhombus

Related topic

ApolloniusTheorem

Related topic

Median

Related topic

ParallelogramLaw2

Generated on Fri Feb 9 21:52:29 2018 by

LaTeXML
Pascal’s mystic hexagram

Pascal’s mystic hexagram

If an

hexagon

A

⁢

D

⁢

B

⁢

F

⁢

C

⁢

E

(not necessarily convex) is

inscribed

into a conic (in particular into a circle), then the points of

intersections

of opposite sides
(

A

⁢

D

with

F

⁢

C

,

D

⁢

B

with

C

⁢

E

and

B

⁢

F

with

E

⁢

A

) are

collinear

. This line is called the

Pascal line

of the hexagon.

A very special case happens when the conic degenerates into two lines, however the

theorem

still holds although this particular case is usually called

Pappus theorem

.

Title

Pascal’s mystic

hexagram

Canonical name

PascalsMysticHexagram

Date of creation

2013-03-22 12:10:49

Last modified on

2013-03-22 12:10:49

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Synonym

Pascal line

Synonym

Pascal’s theorem

Related topic

PappussTheorem

Generated on Fri Feb 9 21:52:31 2018 by

LaTeXML
pedal triangle

pedal triangle

The

pedal triangle

of any

triangle

△

⁢

A

⁢

B

⁢

C

,
is the triangle whose vertices are the feet of

perpendiculars

from

A

,

B

and

C

to their opposite sides in

△

⁢

A

⁢

B

⁢

C

.

In this figure, the

△

⁢

D

⁢

E

⁢

F

is the pedal triangle
of

△

⁢

A

⁢

B

⁢

C

.

In general, for any point

P

inside a triangle, the

pedal triangle

of

P

is a triangle whose vertices are the
feet of perpendiculars from

P

to the sides of the triangle.

In the following figure, the

△

⁢

D

′

⁢

E

′

⁢

F

′

is the pedal
triangle of

P

in

△

⁢

A

⁢

B

⁢

C

.

Title

pedal triangle

Canonical name

PedalTriangle

Date of creation

2013-03-22 13:08:28

Last modified on

2013-03-22 13:08:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51-00

Generated on Fri Feb 9 21:52:32 2018 by

LaTeXML
pentagon

pentagon

A

pentagon

is a 5-sided

planar polygon

.

Regular pentagons

are of particular

interest

for geometers.
On a regular pentagon, the inner angles are equal to

108

∘

.
All ten

diagonals

have the same length. If

s

is the length of a side and

d

is the length of a diagonal, then

d

s

=

1

+

5

2

;

that is, the ratio between a diagonal and a side is the Golden Number.

A regular pentagon (along with its diagonals) can also be obtained as
the projection of a

regular

pentahedron

in four dimensional space
onto a plane determined by two opposite edges.
This is analogous to the way a square with its diagonals can be obtained
as the projection of a tetrahedrononto a plane determined by two opposite edges.

Title

pentagon

Canonical

name

Pentagon

Date of creation

2013-03-22 12:10:19

Last modified on

2013-03-22 12:10:19

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 51-00

Related topic

Triangle

Related topic

Polygon

Related topic

Hexagon

Related topic

RegularDecagonInscribedInCircle

Generated on Fri Feb 9 21:52:34 2018 by

LaTeXML
polygon

polygon

A

polygon

is a simple (

http://planetmath.org/SimpleCurve

),

closed path

that lies on a plane and is composed entirely of

line segments

. In other words, if someone were to walk on a polygon, then the person would end up back where he started; moreover, if a person walks on a polygon so that he travels exactly once around the polygon, then the path (

http://planetmath.org/Path

) never crosses itself, and the person is either walking along a line or turning.

Below are some examples of polygons:

A

side

of a polygon is a line segment on the polygon that is of maximal length. In other words, any line segment that contains a side of a polygon and has a greater length than that side is not entirely on that polygon. A

vertex

of a polygon is an

endpoint

of a side of the polygon. Note that each vertex of a polygon is simultaneously the endpoint of exactly two

adjacent sides

of the polygon.

A polygon with

n

sides is called an

n

-gon

. For small

n

, there are more traditional names:

number of sides

name of polygon

3

triangle

4

quadrilateral

5

pentagon

6

hexagon

7

heptagon

8

octagon

10

decagon

In

spherical geometry

, polygons with only two sides exist. They are called

biangles

.

The

perimeter

of a polygon is the sum of the lengths of its sides.

An

interior angle

of a polygon is the

measure

of an angle formed by two adjacent sides such that the angle is measured with respect to the interior of the polygon. For each polygon in the picture below, the interior angles are marked in blue:

Note that the measure (

http://planetmath.org/AngleMeasure

) of any interior angle of a polygon is

strictly between

0

∘

and

360

∘

and is not equal to

180

∘

.

We have the following criterion for a polygon to be convex:

Theorem

.

A polygon is convex if and only if each of its interior angles has a measure that is strictly less than

180

∘

.

The

angle sum

of a polygon is the sum of the measures of its interior angles. In

Euclidean geometry

, the angle sum of an

n

-gon is exactly

180

⁢

(

n

-

2

)

∘

.

An

exterior angle

of a polygon is any angle that forms a

linear pair

with an interior angle of a polygon. In the picture below, all exterior angles of the triangle are marked in blue:

For a more

rigorous

treatment of polygons, see this entry (

http://planetmath.org/Polygon

).

Title

polygon

Canonical name

Polygon1

Date of creation

2013-03-22 17:35:35

Last modified on

2013-03-22 17:35:35

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

20

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Related topic

Polygon

Related topic

BasicLength

Related topic

Diagonal

Related topic

RegularPolygon

Related topic

Semiperimeter

Related topic

EquilateralPolygon

Related topic

EquiangularPolygon

Defines

side

Defines

vertex

Defines

n

-gon

Defines

n-gon

Defines

perimeter

Defines

interior angle

Defines

angle sum

Defines

exterior angle

Generated on Fri Feb 9 21:52:41 2018 by

LaTeXML
primitive

primitive

A

primitive

is a geometric figure that is left undefined.

Properties

of primitives are assumed via axioms of the

geometry

. From these axioms, one can gain some understanding of primitives, but they are technically not defined.

Primitives are typically the simplest figure considered in the geometry. This occurs because it is easier to create more complicated figures from simpler ones; it seems backwards to leave complicated figures undefined and attempt to define simpler figures from these.

In

Euclidean geometry

, the primitives are typically taken to be point, line, and plane. On PlanetMath, these all have “

definitions

”. Upon investigation, however, Euclid’s definition for point, “that which has no part,” is a description rather than a definition, and the other definitions require knowledge of

vector spaces

, topology, or

point-free geometry

. Similarly, the PlanetMath definitions of line and plane require knowledge of other of mathematics.

The “figure” in the geometric sense is currently a primitive according to PlanetMath. It is used in many different entries, including this one (in the first for example), with the

assumption

that the reader knows what is meant by the .

Title

primitive

Canonical

name

Primitive

Date of creation

2013-03-22 17:11:49

Last modified on

2013-03-22 17:11:49

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Generated on Fri Feb 9 21:52:43 2018 by

LaTeXML