aarjaneiro/planetmath huggingface dataset:
proof of Stewart’s theorem

proof of Stewart’s theorem

Let

θ

be the angle

∠

⁢

A

⁢

X

⁢

B

.

Cosines law

on

△

⁢

A

⁢

X

⁢

B

says

c

2

=

m

2

+

p

2

-

2

⁢

p

⁢

m

⁢

cos

⁡

θ

and thus

cos

⁡

θ

=

m

2

+

p

2

-

c

2

2

⁢

p

⁢

m

Using cosines law on

△

⁢

A

⁢

X

⁢

C

and noting that

ψ

=

∠

⁢

A

⁢

X

⁢

C

=

180

∘

-

θ

and thus

cos

⁡

θ

=

-

cos

⁡

ψ

we get

cos

⁡

θ

=

b

2

-

n

2

-

p

2

2

⁢

p

⁢

n

.

From the

expressions

above we obtain

2

⁢

p

⁢

n

⁢

(

m

2

+

p

2

-

c

2

)

=

2

⁢

p

⁢

m

⁢

(

b

2

-

n

2

-

p

2

)

.

By cancelling

2

⁢

p

on both sides and collecting we are led to

m

2

⁢

n

+

m

⁢

n

2

+

p

2

⁢

n

+

p

2

⁢

m

=

b

2

⁢

m

+

c

2

⁢

n

and from there

m

⁢

n

⁢

(

m

+

n

)

+

p

2

⁢

(

m

+

n

)

=

b

2

⁢

m

+

c

2

⁢

n

. Finally, we note that

a

=

m

+

n

so we conclude that

a

⁢

(

m

⁢

n

+

p

2

)

=

b

2

⁢

m

+

c

2

⁢

n

.

QED

Title

proof of Stewart’s

theorem

Canonical

name

ProofOfStewartsTheorem

Date of creation

2013-03-22 12:38:37

Last modified on

2013-03-22 12:38:37

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 51-00

Related topic

StewartsTheorem

Related topic

ApolloniusTheorem

Related topic

CosinesLaw

Related topic

ProofOfApolloniusTheorem2

Generated on Fri Feb 9 21:53:26 2018 by

LaTeXML
proof of tangents law

proof of tangents law

To prove that

a

-

b

a

+

b

=

tan

⁡

(

A

-

B

2

)

tan

⁡

(

A

+

B

2

)

we start with the

sines law

, which says that

a

sin

⁡

(

A

)

=

b

sin

⁡

(

B

)

.

This implies that

a

⁢

sin

⁡

(

B

)

=

b

⁢

sin

⁡

(

A

)

We can write

sin

⁡

(

A

)

as

sin

⁡

(

A

)

=

sin

⁡

(

A

+

B

2

)

⁢

cos

⁡

(

A

-

B

2

)

+

cos

⁡

(

A

+

B

2

)

⁢

sin

⁡

(

A

-

B

2

)

.

and

sin

⁡

(

B

)

as

sin

⁡

(

B

)

=

sin

⁡

(

A

+

B

2

)

⁢

cos

⁡

(

A

-

B

2

)

-

cos

⁡

(

A

+

B

2

)

⁢

sin

⁡

(

A

-

B

2

)

.

Therefore, we have

a

⁢

(

sin

⁡

(

A

+

B

2

)

⁢

cos

⁡

(

A

-

B

2

)

-

cos

⁡

(

A

+

B

2

)

⁢

sin

⁡

(

A

-

B

2

)

)

=

b

⁢

(

sin

⁡

(

A

+

B

2

)

⁢

cos

⁡

(

A

-

B

2

)

+

cos

⁡

(

A

+

B

2

)

⁢

sin

⁡

(

A

-

B

2

)

)

Dividing both sides by

cos

⁡

(

A

-

B

2

)

⁢

cos

⁡

(

A

+

B

2

)

,

we have,

a

⁢

(

tan

⁡

(

A

+

B

2

)

-

tan

⁡

(

A

-

B

2

)

)

=

b

⁢

(

tan

⁡

(

A

+

B

2

)

+

tan

⁡

(

A

-

B

2

)

)

This gives us

a

b

=

tan

⁡

(

A

+

B

2

)

+

tan

⁡

(

A

-

B

2

)

tan

⁡

(

A

+

B

2

)

-

tan

⁡

(

A

-

B

2

)

Hence we find that

a

-

b

a

+

b

=

a

b

-

1

a

b

+

1

=

tan

⁡

(

A

-

B

2

)

tan

⁡

(

A

+

B

2

)

.

Title

proof of tangents law

Canonical name

ProofOfTangentsLaw

Date of creation

2013-03-22 13:11:04

Last modified on

2013-03-22 13:11:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Proof

Classification

msc 51-00

Generated on Fri Feb 9 21:53:28 2018 by

LaTeXML
proof of Thales’ theorem

proof of Thales’ theorem

Let

M

be the center of the circle through

A

,

B

and

C

.

Then

A

⁢

M

=

B

⁢

M

=

C

⁢

M

and thus the triangles

A

⁢

M

⁢

C

and

B

⁢

M

⁢

C

are isosceles. If

∠

B

M

C

=

:

α

then

∠

⁢

M

⁢

C

⁢

B

=

90

∘

-

α

2

and

∠

⁢

C

⁢

M

⁢

A

=

180

∘

-

α

. Therefore

∠

⁢

A

⁢

C

⁢

M

=

α

2

and

∠

⁢

A

⁢

C

⁢

B

=

∠

⁢

M

⁢

C

⁢

B

+

∠

⁢

A

⁢

C

⁢

M

=

90

∘

.

QED.

Title

proof of Thales’ theorem

Canonical name

ProofOfThalesTheorem

Date of creation

2013-03-22 12:45:27

Last modified on

2013-03-22 12:45:27

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Proof

Classification

msc 51-00

Generated on Fri Feb 9 21:53:30 2018 by

LaTeXML
proof of when is a point inside a triangle

proof of when is a point inside a triangle

Let

𝐮

∈

ℝ

2

,

𝐯

∈

ℝ

2

and

𝟎

∈

ℝ

2

. Let’s consider the convex hull
of the set

T

=

{

𝐮

,

𝐯

,

𝟎

}

.
By

definition

, the convex hull of

T

, noted

c

⁢

o

⁢

T

, is the smallest
convex set that contains

T

. Now, the

triangle

Δ

T

spanned
by

T

is convex and contains

T

. Then

c

⁢

o

⁢

T

⊆

Δ

T

.
Now, every convex

C

set containing

T

must

satisfy

that

t

⁢

𝐮

+

(

1

-

t

)

⁢

𝐯

∈

C

,

t

⁢

𝐮

∈

C

and

t

⁢

𝐯

∈

C

for

0

≤

t

≤

1

(at
least the

convex combination

of the points of

T

are

contained

in

C

). This means that the boundary of

Δ

T

is contained
in

C

. But then every convex combination of points of

∂

⁡

Δ

T

must also be contained in

C

, meaning that

Δ

T

⊆

C

for every convex set containing

T

. In particular,

Δ

T

⊆

c

⁢

o

⁢

T

.

Since the convex hull is exactly the set containing all convex combinations
of points of

T

,

Δ

T

=

c

o

T

=

{

𝐱

∈

ℝ

2

:

𝐱

=

λ

𝐮

+

μ

𝐯

+

(

1

-

λ

-

μ

)

𝟎

,

0

≤

λ

,

μ

,

≤

1

,

0

≤

1

-

λ

-

μ

≤

1

}

we conclude that

𝐱

∈

ℝ

2

is in the triangle
spanned by

T

if and only if

𝐱

=

λ

⁢

𝐮

+

μ

⁢

𝐯

with

0

≤

λ

,

μ

,

≤

1

and

0

≤

λ

+

μ

≤

1

.

Title

proof of when is a point inside a triangle

Canonical

name

ProofOfWhenIsAPointInsideATriangle

Date of creation

2013-03-22 17:57:08

Last modified on

2013-03-22 17:57:08

Owner

joen235 (18354)

Last modified by

joen235 (18354)

Numerical id

4

Author

joen235 (18354)

Entry type

Proof

Classification

msc 51-00

Generated on Fri Feb 9 21:53:32 2018 by

LaTeXML
Ptolemy’s theorem

Ptolemy’s theorem

If

A

⁢

B

⁢

C

⁢

D

is a

cyclic quadrilateral

, then the product of the two

diagonals

is equal to the sum of the products of opposite sides.

A

⁢

C

⋅

B

⁢

D

=

A

⁢

B

⋅

C

⁢

D

+

A

⁢

D

⋅

B

⁢

C

.

When the

quadrilateral

is not cyclic we have the following inequality

A

⁢

B

⋅

C

⁢

D

+

B

⁢

C

⋅

A

⁢

D

>

A

⁢

C

⋅

B

⁢

D

An interesting particular case is when both

A

⁢

C

and

B

⁢

D

are

diameters

, since we get another proof of Pythagoras’ theorem.

Title

Ptolemy’s theorem

Canonical name

PtolemysTheorem

Date of creation

2013-03-22 11:43:13

Last modified on

2013-03-22 11:43:13

Owner

drini (3)

Last modified by

drini (3)

Numerical id

18

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Classification

msc 60K25

Classification

msc 18-00

Classification

msc 68Q70

Classification

msc 37B15

Classification

msc 18-02

Classification

msc 18B20

Related topic

CyclicQuadrilateral

Related topic

ProofOfPtolemysTheorem

Related topic

PtolemysTheorem

Related topic

PythagorasTheorem

Related topic

CrossedQuadrilateral

Generated on Fri Feb 9 21:53:34 2018 by

LaTeXML
Pythagorean theorem

Pythagorean theorem

Pythagorean theorem

states:

If

△

⁢

A

⁢

B

⁢

C

is a

right triangle

, then the

http://planetmath.org/node/1087

square of the length of the

hypothenuse

is equal to the sum of the squares of the two legs. In the following picture, the purple squares add up to the same area as the orange one.

A

⁢

C

2

=

A

⁢

B

2

+

B

⁢

C

2

.

Cosines law

is a

generalization

of Pythagorean theorem for any

triangle

.
It implies that the

converse

of Pythagorean theorem also holds: if
the sides of a triangle

satisfy

a

2

+

b

2

=

c

2

then the triangle is a
right triangle.

Title

Pythagorean theorem

Canonical

name

PythagoreanTheorem

Date of creation

2013-03-22 11:43:09

Last modified on

2013-03-22 11:43:09

Owner

drini (3)

Last modified by

drini (3)

Numerical id

28

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Classification

msc 20G42

Classification

msc 16S40

Classification

msc 57T05

Classification

msc 16W35

Classification

msc 81R50

Classification

msc 16W30

Synonym

Pythagoreas’ theorem

Synonym

Pythagoras theorem

Related topic

Triangle

Related topic

CosinesLaw

Related topic

Hypotenuse

Related topic

RightTriangle

Related topic

PythagoreanTriple

Related topic

ProofOfPythagoreasTheorem

Related topic

PythagorasTheorem

Related topic

PtolemysTheorem

Related topic

FirstPrimitivePythagoreanTriplets

Related topic

PythagoreanTheoremInInnerProductSpaces

Related topic

GeneralizedPythagoreanTheorem

Generated on Fri Feb 9 21:53:35 2018 by

LaTeXML
quadrilateral

quadrilateral

A

quadrilateral

is a four-sided

polygon

.

Very special kinds of quadrilaterals are

parallelograms

(squares,

rhombuses

,

rectangles

, etc.), although

cyclic quadrilaterals

are also interesting on their own.
Notice however, that there are quadrilaterals that are neither parallelograms nor cyclic quadrilaterals.

Title

quadrilateral

Canonical name

Quadrilateral

Date of creation

2013-03-22 12:02:18

Last modified on

2013-03-22 12:02:18

Owner

drini (3)

Last modified by

drini (3)

Numerical id

8

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Related topic

CyclicQuadrilateral

Related topic

Rhombus

Related topic

ParallelogramLaw

Related topic

Rectangle

Related topic

Kite

Related topic

Square

Related topic

Parallelogram

Generated on Fri Feb 9 21:53:37 2018 by

LaTeXML
radius

radius

The

radius

of a circle or sphere is the

distance

from the center of the figure to the outer edge (or surface.)

This

definition

actually holds in

n

dimensions

; so 4th and 5th and

k

-dimensional “spheres” have radii. Since a circle is really a 2-dimensional sphere, its “radius” is merely an instance of the general definition.

Title

radius

Canonical

name

Radius

Date of creation

2013-03-22 12:21:02

Last modified on

2013-03-22 12:21:02

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

5

Author

akrowne (2)

Entry type

Definition

Classification

msc 51-00

Related topic

Diameter2

Generated on Fri Feb 9 21:53:39 2018 by

LaTeXML
rectangle

rectangle

Rectangle

.
A

quadrilateral

whose four angles are equal, that is, whose 4 angles are equal to

90

∘

.

Any rectangle is a

parallelogram

.

This follows from angles

∠

⁢

B

⁢

A

⁢

D

and

∠

⁢

A

⁢

D

⁢

C

adding up

180

∘

.

Since parallelograms have their opposite sides equal, so do rectangles. In the picture,

A

⁢

B

=

C

⁢

D

and

B

⁢

C

=

D

⁢

A

.

Rectangles are the only parallelograms to be also cyclic (since opposite angles add up

180

∘

.

Notice that every square is also a rectangle, but there are rectangles that are not squares

Rectangles have their two

diagonals

equal (since

triangles

A

⁢

B

⁢

C

and

A

⁢

B

⁢

D

are

congruent

), A nice result following from this is that the quadrilateral obtained by joining the

midpoints

of the sides is a

rhombus

.

Since

P

⁢

Q

joins midpoints of sides in triangle

A

⁢

B

⁢

C

, we have

P

⁢

Q

=

C

⁢

A

/

2

. Similarly we have

R

⁢

S

=

C

⁢

A

/

2

,

Q

⁢

R

=

B

⁢

D

/

2

and

S

⁢

P

=

B

⁢

D

/

2

and thus the sides of quadrilateral

P

⁢

Q

⁢

R

⁢

S

are all equal, in other words,

P

⁢

Q

⁢

R

⁢

S

is a rhombus.

Title

rectangle

Canonical

name

Rectangle

Date of creation

2013-03-22 12:02:29

Last modified on

2013-03-22 12:02:29

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Related topic

Quadrilateral

Related topic

Parallelogram

Related topic

Rhombus

Related topic

ParallelogramLaw

Related topic

Square

Generated on Fri Feb 9 21:53:40 2018 by

LaTeXML
regular polyhedron

regular polyhedron

A

regular polyhedron

is a polyhedron such that

•

Every face is a

regular polygon

.

•

On each vertex, the same number of edges concur.

•

The

dihedral angle

between any two faces is always the same.

These polyhedra are also known as

Platonic solids

, since Plato described them in his work. There are only 5 regular polyhedra, as was first shown by Theaetetus, one of Plato’s students. Some sources ascribe to Theaetetus also the discovery of the

dodecahedron

.

The five solids are:

Regular Tetrahedron

It has 6 edges and 4 vertices and 4 faces, each one being an

equilateral triangle

. Its

symmetry group

is

S

4

.

Regular

Hexahedron

Also known as cube. It has 8 vertices, 12 edges and 6 faces each one being a square. Its symmetry group is

S

4

×

C

2

.

Regular Octahedron

It has 6 vertices, 12 edges and 8 faces, each one being an equilateral triangle Its symmetry group is

S

4

×

C

2

.

Regular Dodecahedron

It has 20 vertices, 30 edges and 12 faces, each one being a

regular pentagon

. Its symmetry group is

A

5

×

C

2

.

Regular Icosahedron

It has 12 vertices, 30 edges and 20 faces, each one being an equilateral triangle. Its symmetry group is

A

5

×

C

2

.

Figure 1:

The five Platonic solids – created in Blender 2.36. (Download the

\htmladdnormallink

Blender filehttp://aux.planetmath.org/files/objects/2254/platonic.blend for this picture.)

Note:

A

n

is the

alternating group

of order

n

,

S

n

is the

symmetric group

of order

n

and

C

n

is the

cyclic group

with order

n

.

Title

regular polyhedron

Canonical name

RegularPolyhedron

Date of creation

2013-03-22 12:24:17

Last modified on

2013-03-22 12:24:17

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

20

Author

mathwizard (128)

Entry type

Definition

Classification

msc 51-00

Synonym

Platonic solid

Synonym

regular polyhedra

Synonym

regular

Related topic

RegularPolygon

Related topic

Grafix

Defines

tetrahedron

Defines

octahedron

Defines

dodecahedron

Defines

icosahedron

Defines

regular tetrahedron

Defines

regular octahedron

Defines

regular dodecahedron

Defines

regular icosahedron

Generated on Fri Feb 9 21:53:45 2018 by

LaTeXML
regular triangle

regular triangle

A

regular triangle

is one for which all sides are

congruent

and all

interior angles

are congruent.

All regular triangles are

regular polygons

. Also, by the

isosceles triangle theorem

, the

bisector

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

perpendicular bisector

of the opposite side.

The following statements hold in

Euclidean geometry

for a regular triangle.

•

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

regular triangle

Canonical name

RegularTriangle

Date of creation

2013-03-22 17:12:52

Last modified on

2013-03-22 17:12:52

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Related topic

Triangle

Related topic

IsoscelesTriangle

Related topic

EquilateralTriangle

Related topic

EquiangularTriangle

Related topic

EquivalentConditionsForTriangles

Related topic

RegularPolygon

Generated on Fri Feb 9 21:53:48 2018 by

LaTeXML
rhomboid

rhomboid

A

rhomboid

is a

parallelogram

that is neither a

rhombus

nor a

rectangle

. The word rhomboid comes from the Greek word

ρ

⁢

o

⁢

μ

⁢

β

⁢

o

⁢

ε

⁢

ι

⁢

δ

⁢

η

´

⁢

ς

, which is transliterated as ‘rhomboeidis’. In

The Elements

, Euclid has this definition:

“Of

quadrilateral

figures….a rhomboid (is) that which has its opposite sides and angles equal to each other but is neither

equilateral

(

http://planetmath.org/Equilateral

) nor right-angled (

http://planetmath.org/RightAngle

).”

Below is a picture of a rhomboid.

The for the area of a rhomboid is the same as that for all parallelograms: If

b

is the length of its and

h

is the length of its , then

A

=

b

⁢

h

.

Title

rhomboid

Canonical name

Rhomboid

Date of creation

2013-03-22 17:12:35

Last modified on

2013-03-22 17:12:35

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Related topic

Parallelogram

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

LaTeXML
rhombus

rhombus

A

rhombus

(plural

rhombi

, or

rhombuses

)
is a

quadrilateral

with its four sides equal.
A square is a rhombus,
but a rhombus need not be a square,
since the angles need not all be equal.
Every rhombus is, however, a

parallelogram

.

In any rhombus, the

diagonals

are always

perpendicular

.
A nice result following from this,
is that joining the

midpoints

of the sides, always gives a

rectangle

.

If

D

and

d

are the lengths of the diagonals,
then the area of the rhombus is

D

⁢

d

2

.

Title

rhombus

Canonical name

Rhombus

Date of creation

2013-03-22 12:02:38

Last modified on

2013-03-22 12:02:38

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

14

Author

yark (2760)

Entry type

Definition

Classification

msc 51-00

Related topic

Quadrilateral

Related topic

Parallelogram

Related topic

Rectangle

Related topic

Square

Related topic

ParallelogramLaw

Related topic

Kite

Defines

rhombi

Defines

rhombuses

Generated on Fri Feb 9 21:53:52 2018 by

LaTeXML
right trapezoid

right trapezoid

A

right trapezoid

is a

trapezoid

that has at least two

right angles

. Below is a picture of a right trapezoid.

In some dialects of English (e.g. (

http://planetmath.org/Eg

) British English), this figure is referred to as a

right trapezium

. Because of the modifier “right”, no confusion should arise with this usage.

All

rectangles

are right trapezoids (unless the definition of trapezoid is used, see the entry on trapezoid (

http://planetmath.org/Trapezoid

) for more details). Note also that, in

Euclidean geometry

, a trapezoid cannot have an

odd number

of right angles.

A is a trapezoid that is simultaneously a right trapezoid and an

isosceles trapezoid

. In Euclidean geometry, such trapezoids are automatically rectangles. In

hyperbolic geometry

, such trapezoids are automatically

Saccheri quadrilaterals

. Thus, the phrase “right isosceles trapezoid” occurs rarely.

Right trapezoids are used in the

trapezoidal rule

and

composite trapezoidal rule

for estimating Riemann integrals.

Title

right trapezoid

Canonical name

RightTrapezoid

Date of creation

2013-03-22 17:11:55

Last modified on

2013-03-22 17:11:55

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Synonym

right trapezium

Related topic

LambertQuadrilateral

Related topic

SaccheriQuadrilateral

Generated on Fri Feb 9 21:53:55 2018 by

LaTeXML
right triangle

right triangle

A

triangle

A

⁢

B

⁢

C

is right when one of its angles is equal to

90

∘

(and therefore has two

perpendicular

sides).

The sides of a

right triangle

satisfy the

Pythagorean theorem

.

Title

right triangle

Canonical name

RightTriangle

Date of creation

2013-10-30 8:57:24

Last modified on

2013-10-30 8:57:24

Owner

drini (3)

Last modified by

pahio (2872)

Numerical id

7

Author

drini (2872)

Entry type

Definition

Classification

msc 51-00

Related topic

EquilateralTriangle

Related topic

Triangle

Related topic

Hypotenuse

Related topic

PythagorasTheorem

Related topic

IsoscelesTriangle

Related topic

GeneralizedPythagoreanTheorem

Generated on Fri Feb 9 21:53:56 2018 by

LaTeXML
Rodrigues’ rotation formula

Rodrigues’ rotation formula

Rodrigues’

rotation formula

gives a convenient way to write the general

rotation matrix

in

R

3

.

If

[

v

1

,

v

2

,

v

3

]

is a

unit vector

on the

rotation

axis, and

θ

is the rotation angle about that axis, then the rotation matrix is given by

I

+

sin

⁡

(

θ

)

⁢

A

+

(

1

-

cos

⁡

(

θ

)

)

⁢

A

2

where

I

is the

identity matrix

and

A

=

(

0

-

v

3

v

2

v

3

0

-

v

1

-

v

2

v

1

0

)

.

Title

Rodrigues’ rotation formula

Canonical name

RodriguesRotationFormula

Date of creation

2013-03-22 15:20:04

Last modified on

2013-03-22 15:20:04

Owner

acastaldo (8031)

Last modified by

acastaldo (8031)

Numerical id

7

Author

acastaldo (8031)

Entry type

Result

Classification

msc 51-00

Classification

msc 15-00

Related topic

DecompositionOfOrthogonalOperatorsAsRotationsAndReflections

Generated on Fri Feb 9 21:53:58 2018 by

LaTeXML
scalene triangle

scalene triangle

A

triangle

in which no two sides have the same is called

scalene

.

Below is a comparison of a

scalene triangle

with an

equilateral triangles

and some

isosceles triangles

:

Title

scalene triangle

Canonical name

ScaleneTriangle

Date of creation

2013-03-22 16:05:25

Last modified on

2013-03-22 16:05:25

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Synonym

scalene

Related topic

Triangle

Generated on Fri Feb 9 21:53:59 2018 by

LaTeXML
Schooten theorem

Schooten theorem

Theorem

:

Let

A

⁢

B

⁢

C

be a

equilateral triangle

. If

M

is a
point on the circumscribed circle then the equality

A

⁢

M

=

B

⁢

M

+

C

⁢

M

holds.

Proof:

Let

B

′

∈

(

M

⁢

A

)

so that

M

⁢

B

′

=

B

′

⁢

B

. Because

B

⁢

M

⁢

A

^

=

B

⁢

C

⁢

A

^

=

60

∘

, the

triangle

M

⁢

B

⁢

B

′

is

equilateral

, so

B

⁢

B

′

=

M

⁢

B

=

M

⁢

B

′

. Because

A

⁢

B

=

B

⁢

C

,

B

⁢

B

′

=

B

⁢

M

and

A

⁢

B

⁢

B

′

^

≡

M

⁢

B

⁢

C

^

we have that the triangles

A

⁢

B

⁢

B

′

and

C

⁢

B

⁢

M

are

equivalent

. Since

M

⁢

C

=

A

⁢

B

′

we have that

A

⁢

M

=

A

⁢

B

′

+

B

′

⁢

M

=

M

⁢

C

+

M

⁢

B

.

□

References

1

[Pritchard] Pritchard, Chris (ed.)

The Changing Shape of

Geometry

: Celebrating a Century of Geometry and Geometry Teaching.

Cambridge University Press, 2003.

Title

Schooten theorem

Canonical name

SchootenTheorem

Date of creation

2013-03-22 14:05:50

Last modified on

2013-03-22 14:05:50

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 51-00

Synonym

Ptolemy’s theorem

Generated on Fri Feb 9 21:54:01 2018 by

LaTeXML
sector of a circle

sector of a circle

A sector is a fraction of the interior of a circle, described by a

central angle

θ

.
If

θ

=

2

⁢

π

,

the sector becomes a

complete

circle.

If the central angle is

θ

,

and the radius of the circle is

r

,

then the area of the sector is given by

1

2

⁢

r

2

⁢

θ

This is

obvious

from the fact that the area of a sector is

θ

2

⁢

π

times the area of the circle (which is

π

⁢

r

2

).
Note that, in the formula,

θ

is in radians.

Remark

. Since the length

a

of the arc of the sector is

r

⁢

θ

, the area of the sector is

1

2

⁢

a

⁢

r

, which is equal to the area of a

triangle

with base

=

a

and the height

=

r

.

Title

sector of a circle

Canonical

name

SectorOfACircle

Date of creation

2013-03-22 13:10:20

Last modified on

2013-03-22 13:10:20

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51-00

Synonym

sector

Generated on Fri Feb 9 21:54:03 2018 by

LaTeXML
semiperimeter

semiperimeter

The

semiperimeter

of a

polygon

is half of its

perimeter

(

http://planetmath.org/Perimeter2

).

Semiperimeters are most commonly used for determining the area of a

triangle

. The most famous example of this usage is Heron’s formula. There is also a for the area of a

quadrilateral

which involves semiperimeter.

Title

semiperimeter

Canonical

name

Semiperimeter

Date of creation

2013-03-22 16:58:13

Last modified on

2013-03-22 16:58:13

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Related topic

Perimeter2

Related topic

BasicPolygon

Generated on Fri Feb 9 21:54:05 2018 by

LaTeXML