aarjaneiro/planetmath huggingface dataset:
Simson’s line

Simson’s line

Let

A

⁢

B

⁢

C

a

triangle

and

P

a point on its

circumcircle

(other than

A

,

B

,

C

).
Then the feet of the

perpendiculars

drawn from P to the sides

A

⁢

B

,

B

⁢

C

,

C

⁢

A

(or their prolongations) are

collinear

.

In the picture, the line

passing through

U

,

V

,

W

is a

Simson line

for

△

⁢

A

⁢

B

⁢

C

.

An interesting result form the realm of

analytic geometry

states that the

envelope

formed by Simson’s lines when P varies is a

circular

hypocycloid of three points.

Title

Simson’s line

Canonical name

SimsonsLine

Date of creation

2013-03-22 12:24:34

Last modified on

2013-03-22 12:24:34

Owner

drini (3)

Last modified by

drini (3)

Numerical id

17

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Related topic

Circumcircle

Related topic

Triangle

Generated on Fri Feb 9 21:54:06 2018 by

LaTeXML
sines law

sines law

Sines Law

.

Let

A

⁢

B

⁢

C

be a

triangle

where

a

,

b

,

c

are the sides opposite to

A

,

B

,

C

respectively, and let

R

be the radius of the

circumcircle

. Then the following relation holds:

a

sin

⁡

A

=

b

sin

⁡

B

=

c

sin

⁡

C

=

2

⁢

R

.

Title

sines law

Canonical name

SinesLaw

Date of creation

2013-03-22 11:42:40

Last modified on

2013-03-22 11:42:40

Owner

drini (3)

Last modified by

drini (3)

Numerical id

26

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Classification

msc 97-01

Synonym

law of sines

Related topic

CosinesLaw

Related topic

SinesLawProof

Related topic

Triangle

Generated on Fri Feb 9 21:54:08 2018 by

LaTeXML
sines law proof

sines law proof

Let

A

⁢

B

⁢

C

a

triangle

. Let

T

a point in the

circumcircle

such that

B

⁢

T

is a

diameter

.

So

∠

⁢

A

=

∠

⁢

C

⁢

A

⁢

B

is equal to

∠

⁢

C

⁢

T

⁢

B

(they

subtend

the same arc). Since

△

⁢

C

⁢

B

⁢

T

is a

right triangle

, from the definition of sine we get

sin

⁡

∠

⁢

C

⁢

T

⁢

B

=

B

⁢

C

B

⁢

T

=

a

2

⁢

R

.

On the other hand

∠

⁢

C

⁢

A

⁢

B

=

∠

⁢

C

⁢

T

⁢

B

implies their sines are the same and so

sin

⁡

∠

⁢

C

⁢

A

⁢

B

=

a

2

⁢

R

and therefore

a

sin

⁡

A

=

2

⁢

R

.

Drawing diameters passing by

C

and

A

will let us prove in a

similar

way the relations

b

sin

⁡

B

=

2

⁢

R

and

c

sin

⁡

C

=

2

⁢

R

and we conclude that

a

sin

⁡

A

=

b

sin

⁡

B

=

c

sin

⁡

C

=

2

⁢

R

.

Q.E.D.

Title

sines law proof

Canonical name

SinesLawProof

Date of creation

2013-03-22 11:57:36

Last modified on

2013-03-22 11:57:36

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Proof

Classification

msc 51-00

Related topic

CosinesLaw

Related topic

Triangle

Related topic

SinesLaw

Generated on Fri Feb 9 21:54:10 2018 by

LaTeXML
some proofs for triangle theorems

some proofs for triangle theorems

In the following, only

Euclidean geometry

is considered.

The sum of three angles

A

,

B

, and

C

of a

triangle

is

A

+

B

+

C

=

180

∘

.

The following triangle shows how the angles can be found to make a half revolution, which equals

180

∘

.

□

The area

A

=

r

⁢

s

where

s

is the

semiperimeter

s

=

a

+

b

+

c

2

and

r

is the radius of the inscribed circle can be proven by creating the triangles

△

⁢

B

⁢

A

⁢

O

,

△

⁢

B

⁢

C

⁢

O

, and

△

⁢

A

⁢

C

⁢

O

from the

original triangle

△

⁢

A

⁢

B

⁢

C

, where

O

is the center of the inscribed circle.

A

△

⁢

A

⁢

B

⁢

C

=

A

△

⁢

A

⁢

B

⁢

O

+

A

△

⁢

B

⁢

C

⁢

O

+

A

△

⁢

A

⁢

C

⁢

O

=

r

⁢

c

2

+

r

⁢

a

2

+

r

⁢

b

2

=

r

⁢

(

a

+

b

+

c

)

2

=

r

⁢

s

□

Title

some proofs for triangle theorems

Canonical name

SomeProofsForTriangleTheorems

Date of creation

2013-03-22 14:03:55

Last modified on

2013-03-22 14:03:55

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 51-00

Generated on Fri Feb 9 21:54:16 2018 by

LaTeXML
square

square

A

square

is the

regular

4-gon, that is, a

quadrilateral

whose 4 angles and 4 sides are respectively equal. Each angle of a square must be a

right angle

.
This implies a square is a

parallelogram

that is both a

rhombus

and a

rectangle

at the same time.

Notice, however, that if a quadrilateral has its 4 sides equal, we cannot generally say it is a square, since it might be a rhombus as well.

If

r

is the length of a side, the

diagonals

of a square (which are equal since it’s a rectangle too) have length

r

⁢

2

.

Title

square

Canonical name

Square

Date of creation

2013-03-22 12:02:32

Last modified on

2013-03-22 12:02:32

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Related topic

Quadrilateral

Related topic

Parallelogram

Related topic

Rectangle

Related topic

Rhombus

Related topic

ParallelogramLaw

Related topic

RegularPolygon

Generated on Fri Feb 9 21:54:17 2018 by

LaTeXML
Stewart’s theorem

Stewart’s theorem

Let be given a triangle

△

⁢

A

⁢

B

⁢

C

with

A

⁢

B

=

c

,

B

⁢

C

=

a

,

C

⁢

A

=

b

, and a point

X

on

B

⁢

C

such that

B

⁢

X

=

m

and

X

⁢

C

=

n

. Denote with

p

the length of

A

⁢

X

.

Then

a

⁢

(

p

2

+

m

⁢

n

)

=

b

2

⁢

m

+

c

2

⁢

n

.

Title

Stewart’s theorem

Canonical name

StewartsTheorem

Date of creation

2013-03-22 11:44:07

Last modified on

2013-03-22 11:44:07

Owner

drini (3)

Last modified by

drini (3)

Numerical id

11

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Classification

msc 65-00

Classification

msc 81-00

Related topic

ApolloniusTheorem

Related topic

CosinesLaw

Related topic

ProofOfStewartsTheorem

Related topic

StewartsTheorem

Related topic

ProofOfApolloniusTheorem2

Generated on Fri Feb 9 21:54:19 2018 by

LaTeXML
tangent of circle

tangent of circle

The

tangent

of may be defined by using whatever of the below three

characterisations

.

The is a line

•

whose

distance

from the centre of the circle is equal to the radius of the circle;

•

that is

perpendicular

to a radius of the circle and

passes through

the

end point

of the radius lying on the

circumference

;

•

that has only one common point with the curve.

Note.

The tangent of

ellipse

may also be defined by using the third characterisation.

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Title

tangent of circle

Canonical

name

TangentOfCircle

Date of creation

2013-03-22 17:35:58

Last modified on

2013-03-22 17:35:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 51-00

Classification

msc 51M15

Related topic

TangentOfConicSection

Related topic

ConjugateDiametersOfEllipse

Generated on Fri Feb 9 21:54:28 2018 by

LaTeXML
tangents law

tangents law

Let

A

⁢

B

⁢

C

be a triangle with

a

,

b

and

c

being the sides opposite to

A

,

B

and

C

respectively.
Then the following relation holds.

a

-

b

a

+

b

=

tan

⁡

(

A

-

B

2

)

tan

⁡

(

A

+

B

2

)

.

Title

tangents law

Canonical name

TangentsLaw

Date of creation

2013-03-22 13:11:01

Last modified on

2013-03-22 13:11:01

Owner

giri (919)

Last modified by

giri (919)

Numerical id

5

Author

giri (919)

Entry type

Theorem

Classification

msc 51-00

Synonym

law of tangents

Generated on Fri Feb 9 21:54:30 2018 by

LaTeXML
Thales’ theorem

Thales’ theorem

Let

A

and

B

be two points and

C

a point on the

semicircle

above them. Then the angle

A

⁢

C

⁢

B

is

90

∘

.

Title

Thales’ theorem

Canonical name

ThalesTheorem

Date of creation

2013-03-22 12:45:24

Last modified on

2013-03-22 12:45:24

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 51-00

Related topic

AngleOfViewOfALineSegment

Related topic

ProvingThalesTheoremWithVectors

Related topic

AreaOfSphericalCalotteByMeansOfChord

Generated on Fri Feb 9 21:54:32 2018 by

LaTeXML
trapezoid

trapezoid

A

trapezoid

is a

quadrilateral

with at least one pair of sides

parallel

. Some insist that trapezoids have exactly one pair of sides parallel, in which case

parallelograms

are not trapezoids. Other sources do not restrict the definition in this manner, in which case parallelograms are trapezoids. The convention in PlanetMath is to use the unrestricted definition.

In some dialects of English (e.g. (

http://planetmath.org/Eg

) British English), a trapezoid is referred to as a

trapezium

. Unfortunately, some confusion arises when this word is used, since in other dialects of English (e.g. American English), a

trapezium

is a quadrilateral without any parallel sides.

Below is a picture of a trapezoid.

The

bases

of a trapezoid are its two parallel sides. (If the trapezoid is a parallelogram, either pair of parallel sides can be declared to be its bases.) The

legs

of a trapezoid are the two sides that are not bases. A

height

of a trapezoid is a

line segment

that is

perpendicular

to the bases of the trapezoid and whose

endpoints

lie on the two lines formed by extending the two bases. Typically, heights are drawn so that they

intersect

at least one base of the trapezoid. (For some trapezoids, it is impossible to draw a height that intersects both bases.) Below is a picture of a trapezoid with its bases labelled

b

1

and

b

2

and a height drawn in blue.

The

median

of a trapezoid is the line segment whose endpoints are the

midpoints

of the legs of the trapezoid. Below is a picture of a trapezoid with its median drawn in red.

In the of this entry, only

Euclidean geometry

is considered.

If a trapezoid has bases of lengths

b

1

and

b

2

and a height of length

h

, then the area of the trapezoid is

A

=

1

2

⁢

(

b

1

+

b

2

)

⁢

h

.

Note that the length

m

of the median of a trapezoid is the

arithmetic mean

of the lengths of its bases; i.e. (

http://planetmath.org/Ie

),

m

=

1

2

⁢

(

b

1

+

b

2

)

.

Thus, the area of a trapezoid can also be determined by

A

=

m

⁢

h

.

Title

trapezoid

Canonical name

Trapezoid

Date of creation

2013-03-22 17:11:52

Last modified on

2013-03-22 17:11:52

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Defines

trapezium

Defines

base

Defines

leg

Defines

height

Defines

median

Generated on Fri Feb 9 21:54:34 2018 by

LaTeXML
triangle center

triangle center

On every

triangle

there are points where special lines or circles intersect, and those points usually have very interesting geometrical

properties

. Such points are called

triangle centers

.

Some examples of triangle centers are

incenter

,

orthocenter

,

centroid

,

circumcenter

,

excenters

,

Feuerbach point

,

Fermat points

, etc.

For an online reference please check the

http://faculty.evansville.edu/ck6/tcenters/

Triangle Centers page.

Here is a drawing showing the most important lines and centers of a triangle

(XEukleides

\PMlinktofile

source codetriangulo-rev.euk for the drawing)

Title

triangle center

Canonical

name

TriangleCenter

Date of creation

2013-03-22 11:55:50

Last modified on

2013-03-22 11:55:50

Owner

mps (409)

Last modified by

mps (409)

Numerical id

13

Author

mps (409)

Entry type

Definition

Classification

msc 51-00

Synonym

triangle centre

Synonym

center

Synonym

centre

Related topic

Orthocenter

Related topic

Centroid

Related topic

EulerLine

Related topic

FermatTorricelliTheorem

Generated on Fri Feb 9 21:54:38 2018 by

LaTeXML
triangle solving

triangle solving

Let us consider skew-angled

triangles

.  If one knows three parts of a triangle, among which at least one side, then the other parts may be calculated by using the

law of sines

and the

law of cosines

.  We distinguish four cases:

1.

ASA.

Known two angles and one side, e.g.

α

,

β

,

c

.  Other parts:

γ

=

180

∘

-

(

α

+

β

)

,

a

=

c

⁢

sin

⁡

α

sin

⁡

γ

,

b

=

c

⁢

sin

⁡

β

sin

⁡

γ

Figure 1:

ASA (angle-side-angle)

2.

SSS.

Known all sides

a

,

b

,

c

.  The angles are obtained from

cos

⁡

α

=

b

2

+

c

2

-

a

2

2

⁢

b

⁢

c

,

cos

⁡

β

=

c

2

+

a

2

-

b

2

2

⁢

c

⁢

a

,

cos

⁡

γ

=

a

2

+

b

2

-

c

2

2

⁢

a

⁢

b

.

Figure 2:

SSS (side-side-side)

3.

SAS.

Known two sides and the angle between them, e.g.

b

,

c

,

α

.  Other parts from

a

2

=

b

2

+

c

2

-

2

⁢

b

⁢

c

⁢

cos

⁡

α

,

sin

⁡

β

=

b

⁢

sin

⁡

α

a

,

sin

⁡

γ

=

c

⁢

sin

⁡

α

a

Figure 3:

SAS (side-angle-side)

4.

SSA.

Known two sides and the angle of one of them, e.g.

a

,

b

,

α

.  Other parts are gotten from

sin

⁡

β

=

b

⁢

sin

⁡

α

a

,

γ

=

180

∘

-

(

α

+

β

)

,

c

=

a

⁢

sin

⁡

γ

sin

⁡

α

.

Figure 4:

SSA (side-side-angle)

Since the SSA criterion alone does not prove

congruence

,
it is not surprising that there may not always be a single solution
for

β

here.
In fact, if the first equation gives

sin

⁡

β

>

1

,  then the situation is impossible and the triangle does not exist.  If the equation gives

sin

⁡

β

<

1

,  one gets two distinct values of

β

; an acute

β

1

and an obtuse

β

2

=

180

∘

-

β

1

.  If in this case

β

1

>

α

,  then there are two different triangles as , but if

β

1

≤

α

,  then there is only one triangle.

•

http://svn.gold-saucer.org/repos/PlanetMath/TriangleSolving/triangle.mp

MetaPost source code for the above diagrams

Title

triangle solving

Canonical

name

TriangleSolving

Date of creation

2013-03-22 15:44:02

Last modified on

2013-03-22 15:44:02

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

11

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 51-00

Related topic

Congruence

Generated on Fri Feb 9 21:54:41 2018 by

LaTeXML
unit disc

unit disc

The

open

unit disc

around

P

(where

P

is a given point on the plane), is the set of points whose

distance

from

P

is less than one:

D

1

⁢

(

P

)

=

{

Q

:

|

P

-

Q

|

<

1

}

.

A

closed unit disc

is the set of points whose distance from

P

is less than or equal to one:

D

¯

1

⁢

(

P

)

=

{

Q

:

|

P

-

Q

|

≤

1

}

Unit discs are a special case of unit ball. Without further

specifications

, the term ”unit disc” is used for the open unit disc about the origin,

D

1

⁢

(

0

)

.

The specific set of points in the unit disc, and hence, its visual appearance, depends on the metric being used.

With the

standard metric

, unit discs look like circles of radius one, but changing the metric changes also the corresponding set of points and therefore the shape of the discs.

For

instance

, with the

taxicab metric

discs look like squares, while on the Chebyshev metric a disc is shaped like a rhombus (even though the underlying

topologies

are the same as the

Euclidean

one).

Title

unit disc

Canonical name

UnitDisc

Date of creation

2013-03-22 15:47:01

Last modified on

2013-03-22 15:47:01

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51-00

Classification

msc 54-00

Synonym

unit ball

Related topic

UnitDisk

Related topic

OpenBall

Related topic

Ball

Defines

closed unit disk

Defines

open unit disk

Generated on Fri Feb 9 21:54:42 2018 by

LaTeXML
volume

volume

Note that a

solid

is a three-dimensional figure.

The

volume

of a solid is the amount of space

contained

within the solid. Volume is typically measured in (also called ); i.e. (

http://planetmath.org/Ie

), if the volume of a solid is 5

in

3

, this means that, if five 1 inch by 1 inch by 1 inch cubes are cut appropriately, they can be arranged so that they exactly fill the solid without any overlapping. In

formulas

, volume is almost always denoted using the letter

V

.

All examples provided within this entry are in

Euclidean geometry

.

For certain solids, volume is quite commonly found by multiplying the lengths of three (not necessarily distinct)

line segments

which are related to the solid. Some examples are:

•

cylinders

•

prisms

•

cones

•

pyramids

•

spheres

With the exception of spheres, each of these solids has at least one face that serves as its base. Cylinders and prisms have two bases; for these solids, the height is defined to be the line segment that is

perpendicular

to both bases and whose

endpoints

lie on the bases. If a cylinder or prism has a base of area

B

and a height of length

h

, then its volume is:

V

=

B

⁢

h

Cones and pyramids have one base; for these solids, the height is defined to be the line segment that is perpendicular to the base, has one endpoint on the base, and has the apex of the figure as its other endpoint. If a cone or pyramid has a base of area

B

and a height of length

h

, then its volume is:

V

=

1

3

⁢

B

⁢

h

The volume of a sphere is given by

V

=

4

3

⁢

π

⁢

r

3

.

Remarks

•

Finding the volume of more complicated geometric figures requires knowledge of

calculus

. In fact, the formulas given above can all be derived using calculus.

•

The

concept

of volume is a special case of a more general concept called

measure

, or more appropriately,

product measure

.

Title

volume

Canonical name

Volume

Date of creation

2013-03-22 17:12:58

Last modified on

2013-03-22 17:12:58

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Classification

msc 51M25

Related topic

Area2

Related topic

Prismatoid

Related topic

BasicLength

Related topic

ExampleOfRiemannTripleIntegral

Defines

solid

Generated on Fri Feb 9 21:54:45 2018 by

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when is a point inside a triangle

when is a point inside a triangle

Suppose

u

,

v

are vectors in

ℝ

2

, and

x

=

s

⁢

u

+

t

⁢

v

for some

s

,

t

∈

ℝ

. Then

x

is in the

triangle

spanned by

u

and

v

and the origin if and only if

s

,

t

≥

0

,

s

+

t

≤

1

.

Title

when is a point inside a triangle

Canonical

name

WhenIsAPointInsideATriangle

Date of creation

2013-03-22 15:22:22

Last modified on

2013-03-22 15:22:22

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Theorem

Classification

msc 51-00

Related topic

WhenIsAPointInsideAPolygon

Generated on Fri Feb 9 21:54:46 2018 by

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π and π^2 are irrational

π

and

π

2

are irrational

Theorem 1

.

π

and

π

2

are

irrational

.

Proof.

For any strictly

positive integer

n

,

x

∈

(

0

,

1

)

we define:

f

=

f

⁢

(

x

)

=

x

n

⁢

(

1

-

x

)

n

n

!

=

1

n

!

⁢

∑

m

=

n

2

⁢

n

c

m

⁢

x

m

where

c

m

are

integers

. For

0

<

x

<

1

we have

0

<

f

⁢

(

x

)

<

1

n

!

(1)

For a

contradiction

, suppose

π

2

is

rational

, so that

π

2

=

a

b

, where

a

,

b

are positive integers.

For

x

∈

(

0

,

1

)

let us define

G

⁢

(

x

)

=

b

n

⁢

[

π

2

⁢

n

⁢

f

⁢

(

x

)

-

π

2

⁢

n

-

2

⁢

f

′′

⁢

(

x

)

+

π

2

⁢

n

-

4

⁢

f

(

4

)

⁢

(

x

)

-

…

+

(

-

1

)

n

⁢

f

(

2

⁢

n

)

⁢

(

x

)

]

.

We have that

f

⁢

(

0

)

=

0

and

f

(

m

)

⁢

(

0

)

=

0

if

m

<

n

or

m

>

2

⁢

n

. But, if

n

≤

m

≤

2

⁢

n

, then

f

(

m

)

⁢

(

0

)

=

m

!

n

!

⁢

c

m

,

an integer. Hence

f

⁢

(

x

)

and all its derivates take integral values at

x

=

0

.Since

f

⁢

(

1

-

x

)

=

f

⁢

(

x

)

, the same is true at

x

=

1

so that

G

⁢

(

0

)

and

G

⁢

(

1

)

are integers. We have

d

d

⁢

x

⁢

[

G

′

⁢

(

x

)

⁢

sin

⁡

π

⁢

x

-

π

⁢

G

⁢

(

x

)

⁢

cos

⁡

π

⁢

x

]

=

[

G

′′

⁢

(

x

)

+

π

2

⁢

G

⁢

(

x

)

]

⁢

sin

⁡

π

⁢

x

=

b

n

⁢

π

2

⁢

n

+

2

⁢

f

⁢

(

x

)

⁢

sin

⁡

π

⁢

x

=

π

2

⁢

a

n

⁢

sin

⁡

π

⁢

x

⁢

f

⁢

(

x

)

.

Hence

π

⁢

∫

0

1

a

n

⁢

sin

⁡

π

⁢

x

⁢

f

⁢

(

x

)

⁢

𝑑

x

=

[

G

′

⁢

(

x

)

⁢

sin

⁡

π

⁢

x

π

-

G

⁢

(

x

)

⁢

cos

⁡

π

⁢

x

]

0

1

=

G

⁢

(

0

)

+

G

⁢

(

1

)

,

witch is an integer. But by equation

1

,

0

<

π

⁢

∫

0

1

a

n

⁢

sin

⁡

π

⁢

x

⁢

f

⁢

(

x

)

⁢

𝑑

x

<

π

⁢

a

n

n

!

<

1

.

For a large enough

n

, we obtain a contradiction.

For any integer

n

, if

a

n

is irrational then a is irrational

http://planetmath.org/?op=getobj&from=objects&id=5779

(proof),
and since

π

2

is irrational

π

2

=

π

is also irrational.
∎

The irrationality of

π

was Proved by Lambert in 1761. The above proof is not the original proof due to Lambert.

References

1

G.H.Hardy and E.M.Wright

An

Introduction

to the Theory of
Numbers

, Oxford University Press, 1959

See also

•

The MacTutor History of Mathematics Archive,

http://www-gap.dcs.st-and.ac.uk/ history/HistTopics/Pi_through_the_ages.html

A history of
Pi

•

The MacTutor History of Mathematics Archive,

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Lambert.html

Johann Heinrich
Lambert

•

http://numbers.computation.free.fr/

Constants

/Miscellaneous/irrationality.html

Irrationality proofs

Title

π

and

π

2

are irrational

Canonical name

piAndpi2AreIrrational

Date of creation

2013-03-22 14:44:00

Last modified on

2013-03-22 14:44:00

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

15

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 51-00

Classification

msc 11-00

Generated on Fri Feb 9 21:54:49 2018 by

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adjacent

adjacent

Given a

right triangle

with an

acute angle

θ

, the side of the triangle that is

adjacent

to

θ

is the side of the triangle that is also a side (

http://planetmath.org/Angle

) of

θ

and is not the

hypotenuse

.

When a phrase such as “adjacent of an angle” is used, one must determine from context whether it refers to this

definition

of adjacent or the other definition of adjacent (

http://planetmath.org/Adjacent3

). Note that the definition supplied above is specifically for right triangles.

Title

adjacent

Canonical

name

Adjacent

Date of creation

2013-03-22 15:59:26

Last modified on

2013-03-22 15:59:26

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

17

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-01

Synonym

adjacent side

Related topic

Sohcahtoa

Generated on Fri Feb 9 21:54:51 2018 by

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