aarjaneiro/planetmath huggingface dataset:
axiomatic geometry

axiomatic geometry

Axiomatic geometry

can be traced back to the time of Euclid. In his book

Elements

, written back in the 300’s B.C., Euclid gave five rules, or

postulates

, describing how
points, lines,

line segments

, etc behave as they are ordinarily
perceived. Based on these postulates, he set out to prove hundreds of
properties. Today, these properties are under the field of study known as

plane

Euclidean geometry

, more popularly known as high school

geometry

.
The systematic and axiomatic approach to proving geometric facts is what makes his

Elements

one of the most important contributions to mathematics.

One key feature of Euclid’s axioms is the abundance of what are known today as
the

undefined terms

. Geometric notions such as points, lines, and
circles are mentioned in his axioms but never clearly defined. For example, Euclid
called a “point” as “that which has no part”. But what the meaning of “part” was
never clarified. It is because of this abundance of undefined terms, Euclid’s postulates by
today’s mathematical standards lack rigor. While some undefined terms give no serious
problems, others create holes in proofs which are unacceptable. In the late 19th century,
David Hilbert published his classic

Foundations of Geometry

, putting Euclid’s postulates
on a more solid ground. In the book, he broke down Euclid’s postulates into five
groups of axioms:

1.

incidence axioms

2.

order axioms

3.

congruence axioms

4.

continuity axiom

5.

parallel axiom (

http://planetmath.org/ParallelPostulate

)

These axioms have been shown to be

independent

of each other, in the sense that no one axiom can be
proved from the rest, and

consistent

, in the sense that no

contradictions

can be derived from them. These
axioms today serve as the

foundation

of plane Euclidean geometry.

Since Hilbert’s work, the natural next step is to look at geometries that are not

Euclidean

. In other words,
geometric models that lack one or several of the “Euclidean axioms” above. The result has been the many
exotic geometries that, surprisingly, have found applications in other places.

Title

axiomatic geometry

Canonical name

AxiomaticGeometry

Date of creation

2013-03-22 18:00:27

Last modified on

2013-03-22 18:00:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Topic

Classification

msc 51-01

Classification

msc 51-00

Related topic

EuclideanAxiomByHilbert

Generated on Fri Feb 9 21:55:10 2018 by

LaTeXML
calculator trigonometric functions

calculator trigonometric functions

The

calculator trigonometric functions

are the basic

trigonometric functions

that appear on scientific and graphing calculators. These

functions

are

sin

,

cos

, and

tan

. Knowing which of the trigonometric functions are the calculator trigonometric functions is useful for

determining signs of trigonometric functions

.

Title

calculator trigonometric functions

Canonical name

CalculatorTrigonometricFunctions

Date of creation

2013-03-22 16:06:14

Last modified on

2013-03-22 16:06:14

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

5

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-01

Classification

msc 97D40

Related topic

DeterminingSignsOfTrigonometricFunctions

Related topic

Trigonometry

Generated on Fri Feb 9 21:55:11 2018 by

LaTeXML
coordinate systems

coordinate systems

Plane

1.

Cartesian coordinates

2.

Polar coordinates

ℝ

3

1.

Cartesian coordinates (

http://planetmath.org/CartesianCoordinates

)

2.

Spherical coordinates

3.

Cylindrical coordinates

Sphere

S

2

1.

Stereographic projection

2.

Mercator projection

3.

..

Miscellaneous

1.

Barycentric coordinates

2.

Homogeneous coordinates

(

x

,

y

,

t

)

in plane,

(

x

,

y

,

z

,

t

)

in

ℝ

3

Differential geometry

1.

Normal coordinates (

http://planetmath.org/RiemannNormalCoordinates

) in Riemann geometry

2.

Darboux coordinates

(

http://planetmath.org/DarbouxsTheoremSymplecticGeometry

) in

symplectic geometry

3.

Gaussian coordinates

4.

Fermi coordinates for a

submanifold

5.

Boundary

coordinates

Coordinate transformations

1.

affine transformations

(

http://planetmath.org/AffineTransformation

)

2.

continuous transformations (

http://planetmath.org/HomotopyOfPaths

) in

algebraic topology

(

homotopy theory

)

3.

Lorentz group

transformations

4.

Cartesian-to-polar coordinate transformations

5.

Cartesian-to-spherical coordinate transformations

6.

A.

7.

B.

Title

coordinate systems

Canonical name

CoordinateSystems

Date of creation

2013-03-22 15:12:53

Last modified on

2013-03-22 15:12:53

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

16

Author

matte (1858)

Entry type

Definition

Classification

msc 51-01

Related topic

StereographicProjection

Generated on Fri Feb 9 21:55:24 2018 by

LaTeXML
Euclidean geometry of plane

Euclidean geometry of plane

Basic objects

1.

point,

2.

line, ray,

line segment

,

3.

plane,

4.

angle,

5.

convex angle

,

6.

circle,

7.

tangent line

,

8.

secant line

,

9.

normal line

10.

center normal

11.

angle bisector

12.

triangle

,

13.

congruent

triangles (

http://

planetmath

.org/

Congruence

),

14.

triangle solving

,

15.

polygon

,

16.

regular polygon

Basic concepts

1.

incidence (

http://planetmath.org/IncidenceGeometry

),

2.

betweenness,

3.

congruence (

http://planetmath.org/CongruenceAxioms

),

4.

similarity

,

5.

continuity,

6.

parallelism

,

7.

perpendicularity

,

8.

corresponding angles in

transversal

cutting

9.

intercept theorem

10.

locus

11.

dilation

and

translation

,

12.

rotation

and

reflection

,

13.

inversion of plane

,

14.

harmonic division

,

15.

angle of view

,

16.

compass and straightedge constructions

Analytic geometry

1.

Cartesian coordinates

,

2.

polar coordinates

,

3.

line in plane

4.

slope,

slope angle

,

condition of orthogonality

,

5.

parabola

,

6.

circle (

http://planetmath.org/Circle

),

7.

ellipse

,

8.

hyperbola

,

9.

conjugate hyperbola

,

10.

unit hyperbola

,

11.

tangent of conic section

,

12.

hyperbolas orthogonal to ellipses

13.

power of point

,

14.

chordal

,

15.

confocal

,

16.

transition to skew-angled coordinates

,

17.

conjugate diameters of ellipse

,

18.

perimeter of ellipse

Title

Euclidean geometry of plane

Canonical name

EuclideanGeometryOfPlane

Date of creation

2013-03-22 15:12:13

Last modified on

2013-03-22 15:12:13

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

30

Author

matte (1858)

Entry type

Topic

Classification

msc 51-01

Synonym

plane geometry

Related topic

EuclideanGeometryOfTheSpace

Related topic

AnalyticGeometry

Generated on Fri Feb 9 21:55:28 2018 by

LaTeXML
Euclidean geometry of space

Euclidean geometry of space

Please help to list the existing and future entries on spatial

geometry

!

Basic objects

1.

point

2.

line, ray,

line segment

3.

plane

4.

normal plane

5.

dihedral angle

6.

cylinder

7.

cone

8.

truncated cone

9.

polyhedron

10.

cuboid

11.

sphere

12.

parts of a ball

13.

great circle

14.

surface

15.

solid angle

16.

Plato’s mathematics

Basic concepts and quantities

1.

congruence (

http://planetmath.org/Congruence

)

2.

similarity in geometry

3.

distance of non-parallel lines

4.

angle between two lines

5.

angle between line and plane

6.

normal of plane

7.

parallelism of line and plane

8.

parallelism of two planes

9.

angle between two planes

10.

perpendicular planes

11.

area of a

spherical triangle

12.

area of spherical zone

13.

volume of spherical cap and spherical sector

14.

volume of solid of revolution

15.

Gaussian curvature

16.

mean curvature at surface point

17.

curvature

(

space curve

)

18.

Torricelli’s trumpet

Analytic geometry

1.

coordinate way

2.

direction cosines

3.

equation of plane

4.

line in space

5.

quadratic surfaces

6.

tangent plane of quadratic surface

7.

generatrices of one-sheeted hyperboloid

8.

generatrices of hyperbolic paraboloid

9.

ruled surface

10.

developable surface

11.

surface of revolution

12.

torus

13.

conic section

Title

Euclidean geometry of space

Canonical name

EuclideanGeometryOfSpace

Date of creation

2013-03-22 17:32:00

Last modified on

2013-03-22 17:32:00

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Topic

Classification

msc 51-01

Synonym

space geometry

Related topic

EuclideanGeometryOfThePlane

Related topic

AnalyticGeometry

Generated on Fri Feb 9 21:55:29 2018 by

LaTeXML
intercept theorem

intercept theorem

Theorem

.

If two

intersecting lines

are cut by

parallel lines

, the

line segments

cut by the parallel lines from one of the lines are proportional to the corresponding line segments cut by them from the other line.

The theorem may be condensed to the following form:

•

If a line

parallel

to a side (

http://planetmath.org/

Triangle

)

B

⁢

C

of a triangle

A

⁢

B

⁢

C

intersects

the other sides in the points

D

and

E

, then the

proportion equation

B

⁢

D

:

D

⁢

A

=

C

⁢

E

:

E

⁢

A

(1)

is true.

The

intercept theorem

has been known by the ancient Babylonians and Egyptians, but the first known proof is found in Euclid’s

Elements

.

Proof.

The

areas of triangles

, which have equal heights, are proportional to the bases of the triangles; if the bases are equal, then also the areas are equal.  These facts are used in the

B

⁢

D

:

D

⁢

A

=

Δ

⁢

B

⁢

D

⁢

E

:

Δ

⁢

D

⁢

A

⁢

E

=

Δ

⁢

C

⁢

E

⁢

D

:

Δ

⁢

E

⁢

A

⁢

D

=

C

⁢

E

:

E

⁢

A

of equalities. Q.E.D.

Title

intercept theorem

Canonical name

InterceptTheorem

Date of creation

2013-03-22 18:49:42

Last modified on

2013-03-22 18:49:42

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 51-01

Classification

msc 51M04

Related topic

AreaOfAPolygonalRegion

Related topic

SimilarTriangles

Related topic

MidSegmentTheorem

Generated on Fri Feb 9 21:55:48 2018 by

LaTeXML
parallellism in Euclidean plane

parallellism in Euclidean plane

Two distinct lines in the

Euclidean plane

are

parallel

to each other if and only if they do not

intersect

, i.e. (

http://planetmath.org/Ie

) if they have no common point. By convention, a line is parallel to itself.

The

parallelism

of

l

and

m

is denoted

l

∥

m

.

Parallelism is an

equivalence relation

on the set of the lines of the plane. Moreover, two nonvertical lines are parallel if and only if they have the same slope. Thus, slope is a natural way of determining the

equivalence classes

of lines of the plane.

Title

parallellism in Euclidean plane

Canonical name

ParallellismInEuclideanPlane

Date of creation

2013-03-22 17:12:38

Last modified on

2013-03-22 17:12:38

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 51-01

Synonym

parallelism

Synonym

parallelism in plane

Synonym

parallelism of lines

Related topic

Slope

Related topic

ParallelPostulate

Related topic

ParallelCurve

Related topic

PerpendicularityInEuclideanPlane

Defines

parallel

Defines

parallel lines

Defines

parallelism

Generated on Fri Feb 9 21:55:50 2018 by

LaTeXML
perpendicularity in Euclidean plane

perpendicularity in Euclidean plane

Two lines in the

Euclidean plane

are

perpendicular

to each other if and only if they intersect and two of the angles they form are

congruent

.

This definition on the one in Hilbert’s

Grundlagen der Geometrie

(“Ein Winkel, welcher einem seiner Nebenwinkel kongruent ist, heißt ein

rechter Winkel

”).

The

perpendicularity

of

l

and

m

is denoted

l

⊥

m

.

References

1

D. Hilbert

:

Grundlagen der Geometrie

. Neunte Auflage, revidiert und ergänzt von Paul Bernays.  B. G. Teubner Verlagsgesellschaft, Stuttgart (1962).

Title

perpendicularity in Euclidean plane

Canonical name

PerpendicularityInEuclideanPlane

Date of creation

2013-04-19 15:00:12

Last modified on

2013-04-19 15:00:12

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 51-01

Related topic

ConditionOfOrthogonality

Related topic

MutualPositionsOfVectors

Related topic

AngleBetweenTwoLines

Related topic

ParallellismInEuclideanPlane

Related topic

OrthogonalCircles

Related topic

DihedralAngle

Defines

perpendicularity

Defines

perpendicular

Defines

orthogonality

Defines

orthogonal

Generated on Fri Feb 9 21:55:51 2018 by

LaTeXML
polar tangential angle

polar tangential angle

The angle, under which a

polar curve

is by a line through the origin, is called the

polar tangential angle

belonging to the

intersection

point on the curve.

Given a polar curve

r

=

r

⁢

(

φ

)

(1)

in

polar coordinates

r

,

φ

,  we derive an expression for the

tangent

of the polar tangential angle

ψ

, using the classical

differential

geometric method.

The point

P

of the curve given by (1) corresponds to the

polar angle

φ

=

∠

⁢

P

⁢

O

⁢

A

and the polar radius

r

=

O

⁢

P

.  The “near” point

P

′

corresponds to the polar angle

φ

+

d

⁢

φ

=

∠

⁢

P

′

⁢

O

⁢

A

and the polar radius

r

+

d

⁢

r

=

O

⁢

P

′

.  In the diagram,

P

′

⁢

Q

is the arc of the circle with

O

as centre and

O

⁢

P

′

as radius.  Thus, in the triangle-like figure

P

⁢

P

′

⁢

Q

we have

P

′

⁢

Q

P

⁢

Q

=

(

r

+

d

⁢

r

)

⁢

d

⁢

φ

d

⁢

r

=

r

+

d

⁢

r

d

⁢

r

d

⁢

φ

.

(2)

This figure can be regarded as an

infinitesimal

right triangle

with the

catheti

P

′

⁢

Q

and

P

⁢

Q

.  Accordingly, their ratio (2) is the tangent of the

acute angle

P

of the

triangle

.  Because the addend

d

⁢

r

in the last numerator in negligible compared with the addend

r

, it can be omitted.  Hence we get the tangent

tan

⁡

ψ

=

r

d

⁢

r

d

⁢

φ

of the polar tangential angle, i.e.

tan

⁡

ψ

=

r

⁢

(

φ

)

r

′

⁢

(

φ

)

.

(3)

Title

polar tangential angle

Canonical name

PolarTangentialAngle

Date of creation

2013-03-22 19:02:32

Last modified on

2013-03-22 19:02:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Derivation

Classification

msc 51-01

Classification

msc 53A04

Related topic

LogarithmicSpiral

Generated on Fri Feb 9 21:56:04 2018 by

LaTeXML
terminal ray

terminal ray

Let an angle whose in radians is

θ

be placed the

Cartesian plane

such that one of its rays

R

1

corresponds to the nonnegative

x

axis and one can go from the point

(

1

,

0

)

to the point that is the

intersection

of the other ray

R

2

of the angle with the circle

x

2

+

y

2

=

1

by traveling exactly

θ

units on the circle. (If

θ

is

positive

, the

distance

should be traveled counterclockwise; otherwise, the distance

|

θ

|

should be traveled clockwise. Also, note that “other ray” is used quite loosely, as it may also correspond to the nonnegative

x

axis also.) Then

R

2

is the

terminal ray

of the angle.

The picture below shows the terminal ray

R

2

of the angle

θ

=

2

⁢

π

3

.

Title

terminal ray

Canonical

name

TerminalRay

Date of creation

2013-03-22 16:06:11

Last modified on

2013-03-22 16:06:11

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

12

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-01

Related topic

Trigonometry

Related topic

CyclometricFunctions

Generated on Fri Feb 9 21:56:06 2018 by

LaTeXML
aspect ratio

aspect ratio

The

aspect ratio

of a rectangular image display area is the ratio of the width of the

rectangle

to its height, expressed either as a ratio of coprime integers or a real number to 1. The two most commonly used aspect ratios are “fullscreen” (4:3 or 1.66:1) and “widescreen” (16:9 OR 2.35:1).

For example, a detail from Leonardo da Vinci’s

Mona Lisa

in a fullscreen aspect ration:

And in a widescreen aspect ratio:

Title

aspect ratio

Canonical

name

AspectRatio

Date of creation

2013-03-22 18:30:19

Last modified on

2013-03-22 18:30:19

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51A05

Classification

msc 00A06

Generated on Fri Feb 9 21:56:07 2018 by

LaTeXML
bisectors theorem

bisectors theorem

Bisectors theorem

.

Let

A

⁢

B

⁢

C

a

triangle

, and

A

⁢

P

the

angle bisector

of

∠

⁢

C

⁢

A

⁢

B

. Then

B

⁢

P

P

⁢

C

=

B

⁢

A

C

⁢

A

.

Generalization of bisectors theorem

.

Let

A

⁢

B

⁢

C

a triangle, and

P

any point on the side

B

⁢

C

. Then

B

⁢

P

P

⁢

C

=

B

⁢

A

⁢

sin

⁡

B

⁢

A

⁢

P

C

⁢

A

⁢

sin

⁡

P

⁢

A

⁢

C

.

The latest theorem even holds when

P

is not on the segment

B

⁢

C

but anywhere on the line (possibly at the infnity), but in this case, the ratio

B

⁢

P

/

P

⁢

C

must be done with

directed segments

and the angles considered as

directed angles

.

Title

bisectors theorem

Canonical name

BisectorsTheorem

Date of creation

2013-03-22 14:49:23

Last modified on

2013-03-22 14:49:23

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Theorem

Classification

msc 51A05

Synonym

angle bisector theorem

Synonym

generalization of bisectors theorem

Related topic

CevasTheorem

Related topic

HarmonicDivision

Generated on Fri Feb 9 21:56:09 2018 by

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Ceva’s theorem

Ceva’s theorem

Let

A

⁢

B

⁢

C

be a given

triangle

and

P

any point of the plane. If

X

is the

intersection

point of

A

⁢

P

with

B

⁢

C

,

Y

the intersection point of

B

⁢

P

with

C

⁢

A

and

Z

is the intersection point of

C

⁢

P

with

A

⁢

B

, then

A

⁢

Z

Z

⁢

B

⋅

B

⁢

X

X

⁢

C

⋅

C

⁢

Y

Y

⁢

A

=

1

.

Conversely

, if

X

,

Y

,

Z

are points on

B

⁢

C

,

C

⁢

A

,

A

⁢

B

respectively, and if

A

⁢

Z

Z

⁢

B

⋅

B

⁢

X

X

⁢

C

⋅

C

⁢

Y

Y

⁢

A

=

1

then

A

⁢

X

,

B

⁢

Y

,

C

⁢

Z

are

concurrent

.

Remarks: All the segments are

directed segments

(that is

A

⁢

B

=

-

B

⁢

A

), and so

theorem

is valid even if the points

X

,

Y

,

Z

are in the prolongations (even at the

infinity

) and

P

is any point on the plane (or at the infinity).

Title

Ceva’s theorem

Canonical name

CevasTheorem

Date of creation

2013-03-22 11:57:15

Last modified on

2013-03-22 11:57:15

Owner

drini (3)

Last modified by

drini (3)

Numerical id

16

Author

drini (3)

Entry type

Theorem

Classification

msc 51A05

Related topic

Triangle

Related topic

Median

Related topic

Centroid

Related topic

Orthocenter

Related topic

OrthicTriangle

Related topic

Cevian

Related topic

Incenter

Related topic

GergonnePoint

Related topic

MenelausTheorem

Related topic

ProofOfVanAubelTheorem

Related topic

VanAubelTheorem

Related topic

BisectorsTheorem

Related topic

DirectedSegment

Generated on Fri Feb 9 21:56:10 2018 by

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cotriangular space

cotriangular space

A

cotriangular space

is a

http://planetmath.org/node/PartialLinearSpace

partial linear space

(

P

,

L

)

in which

•

each line contains exactly three points, and

•

given a point not
on a line, the point is

collinear

with zero or two points of the line.

Title

cotriangular space

Canonical

name

CotriangularSpace

Date of creation

2013-03-22 17:15:34

Last modified on

2013-03-22 17:15:34

Owner

pan (17366)

Last modified by

pan (17366)

Numerical id

9

Author

pan (17366)

Entry type

Definition

Classification

msc 51A05

Generated on Fri Feb 9 21:56:12 2018 by

LaTeXML
Hesse configuration

Hesse configuration

A

Hesse configuration

is a set

P

of nine non-collinear points in the

projective plane

over a field

K

such that any line through two points of

P

contains exactly three points of

P

. Then there are 12 such lines through

P

. A Hesse configuration exists if and only if the field

K

contains a

primitive

third root of unity. For such

K

the

projective automorphism

group

PGL

⁢

(

3

,

K

)

acts transitively on all possible Hesse configurations.

The

configuration

P

with its intersection structure of 12 lines is

isomorphic

to the

affine space

A

=

𝔽

2

where

𝔽

is a field with three

elements

.

The group

Γ

⊂

PGL

⁢

(

3

,

K

)

of all

symmetries

that map

P

onto itself has order 216 and it is isomorphic to the group of

affine transformations

of

A

that have

determinant

1. The

stabilizer

in

Γ

of any of the 12 lines through

P

is a

cyclic subgroup

of order three and

Γ

is generated by these

subgroups

.

The

symmetry group

Γ

is isomorphic to

G

⁢

(

K

)

/

Z

⁢

(

K

)

where

G

⁢

(

K

)

⊂

GL

⁢

(

3

,

K

)

is a group of order 648 generated by

reflections

of order three and

Z

⁢

(

K

)

is its cyclic center of order three. The reflection group

G

⁢

(

ℂ

)

is called the Hesse group which appears as

G

25

in the

classification

of finite complex reflection groups by Shephard and Todd.

If

K

is

algebraically closed

and the

characteristic

of

K

is not 2 or 3 then the nine inflection points of an

elliptic curve

E

over

K

form a Hesse configuration.

Title

Hesse configuration

Canonical name

HesseConfiguration

Date of creation

2013-03-22 14:04:04

Last modified on

2013-03-22 14:04:04

Owner

debosberg (3620)

Last modified by

debosberg (3620)

Numerical id

8

Author

debosberg (3620)

Entry type

Definition

Classification

msc 51A05

Classification

msc 51A45

Classification

msc 51E20

Related topic

ProjectiveSpace

Related topic

AffineSpace

Related topic

EllipticCurve

Generated on Fri Feb 9 21:56:14 2018 by

LaTeXML
Menelaus’ theorem

Menelaus’ theorem

If the points

X

,

Y

and

Z

are on the sides of a

triangle

A

⁢

B

⁢

C

(including their prolongations),

collinear

and do not coincide with any of the points

A

,

B

and

C

, then the equation

A

⁢

Z

Z

⁢

B

⋅

B

⁢

Y

Y

⁢

C

⋅

C

⁢

X

X

⁢

A

=

-

1

holds (all segments are

directed line segments

). The converse of this theorem also holds (thus: three points on the prolongations of the triangle’s sides are collinear if the above equation holds).

Title

Menelaus’ theorem

Canonical name

MenelausTheorem

Date of creation

2013-03-22 12:46:20

Last modified on

2013-03-22 12:46:20

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

7

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 51A05

Related topic

CevasTheorem

Related topic

TrigonometricVersionOfCevasTheorem

Related topic

Collinear

Generated on Fri Feb 9 21:56:15 2018 by

LaTeXML