aarjaneiro/planetmath huggingface dataset:
fundamental theorem on isogonal lines

fundamental theorem on isogonal lines

Let

△

⁢

A

⁢

B

⁢

C

be a

triangle

and

A

⁢

X

,

B

⁢

Y

,

C

⁢

Z

three concurrent lines at

P

.
If

A

⁢

X

′

,

B

⁢

Y

′

,

C

⁢

Z

′

are the respective

isogonal conjugate

lines for

A

⁢

X

,

B

⁢

Y

,

C

⁢

Z

, then

A

⁢

X

′

,

B

⁢

Y

′

,

C

⁢

Z

′

are also

concurrent

at some point

P

′

.

An applications of this

theorem

proves the

existence

of

Lemoine point

(for it is the

intersection

point of the

symmedians

):

This theorem is a direct

consequence

of Ceva’s theorem (trigonometric version).

Title

fundamental theorem on isogonal lines

Canonical name

FundamentalTheoremOnIsogonalLines

Date of creation

2013-03-22 13:01:16

Last modified on

2013-03-22 13:01:16

Owner

drini (3)

Last modified by

drini (3)

Numerical id

4

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Related topic

Isogonal

Related topic

IsogonalConjugate

Related topic

LemoinePoint

Related topic

Symmedian

Related topic

Triangle

Generated on Fri Feb 9 21:51:09 2018 by

LaTeXML
Garfield’s proof of Pythagorean theorem

Garfield’s proof of Pythagorean theorem

James Garfield, the

20

th

president of the

United States

, gave the following proof of the

Pythagorean Theorem

in 1876. Consider the following

trapezoid

(note that this picture is half of the diagram used in Bhaskara’s proof of the Pythagorean theorem (

http://

planetmath

.org/ProofOfPythagoreasTheorem

)).

Recall that the area of a trapezoid with two

parallel

sides (in this case, the left and right sides)

s

1

and

s

2

and height

h

is

h

⁢

s

1

+

s

2

2

So the area of the trapezoid above is

(

a

+

b

)

⁢

a

+

b

2

=

(

a

+

b

)

2

2

The area of the yellow

triangle

(and that of the blue triangle) is

a

⁢

b

2

while the area of the red triangle (also a

right triangle

) is

c

2

2

The two areas must be equal, so

(

a

+

b

)

2

2

=

2

⁢

a

⁢

b

2

+

c

2

2

a

2

+

2

⁢

a

⁢

b

+

b

2

2

=

a

⁢

b

+

c

2

2

a

2

+

2

⁢

a

⁢

b

+

b

2

=

2

⁢

a

⁢

b

+

c

2

a

2

+

b

2

=

c

2

Title

Garfield’s

proof of Pythagorean theorem

Canonical name

GarfieldsProofOfPythagoreanTheorem

Date of creation

2013-03-22 17:09:33

Last modified on

2013-03-22 17:09:33

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

10

Author

rm50 (10146)

Entry type

Proof

Classification

msc 51-00

Generated on Fri Feb 9 21:51:18 2018 by

LaTeXML
generalized Pythagorean theorem

generalized Pythagorean theorem

Theorem 1

If three

similar

polygons

are constructed on the sides
of a

right triangle

, then the area of the polygon constructed on the

hypotenuse

is
equal to the sum of the areas of the polygons constructed on the legs.

We when say that the polygon is constructed on a side of the right triangle, we mean
that the polygon shares an entire side with the polygon.

Beginning of proof.
First, it suffices to prove the theorem for polygons of only one shape.
Suppose that the areas of two polygons

P

and

P

′

of different shapes
constructed on some side of the triangle have a ratio

k

. Then the

areas
of polygons

similar to them (say

R

and

R

′

) and constructed on another side
which is

m

times longer, will be

m

2

times larger for both shapes. Therefore,
they will have the same ratio,

k

. Hence if the areas of

P

′

,

Q

′

,

R

′

satisfy
the property that the first two add up to the third one, then the same
will hold true for the areas of

P

,

Q

and

R

where are

k

times greater.

So instead of constructing a square on each side, as Euclid did, we use a right
triangle that is similar to the original right triangle. And instead of constructing
the triangle on the outside, we use the inside of the triangle.

Drop an

altitude

of the right triangle to its hypotenuse. This altitude divides the
triangle into two triangles and each is similar to the

original triangle

.
We now have three similar right triangles constructed on the sides of the original
right triangle, and two of them add up to the third one.

End of proof.

Title

generalized Pythagorean theorem

Canonical name

GeneralizedPythagoreanTheorem

Date of creation

2013-03-22 17:13:58

Last modified on

2013-03-22 17:13:58

Owner

yogis (15158)

Last modified by

yogis (15158)

Numerical id

12

Author

yogis (15158)

Entry type

Theorem

Classification

msc 51-00

Synonym

Pythagorean theorem

Related topic

RightTriangle

Related topic

Polygon

Related topic

PythagorasTheorem

Related topic

ProofOfPythagoreanTheorem2

Generated on Fri Feb 9 21:51:20 2018 by

LaTeXML
Gergonne point

Gergonne point

Let

A

⁢

B

⁢

C

be a

triangle

and

D

,

E

,

F

where the

incircle

touches the sides

B

⁢

C

,

C

⁢

A

,

A

⁢

B

respectively. Then the lines

A

⁢

D

,

B

⁢

E

,

C

⁢

F

are

concurrent

, and the common point is called the

Gergonne point

of the triangle

.

Title

Gergonne point

Canonical name

GergonnePoint

Date of creation

2013-03-22 12:10:52

Last modified on

2013-03-22 12:10:52

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Related topic

CevasTheorem

Related topic

Triangle

Related topic

LemoinePoint

Related topic

GergonneTriangle

Related topic

Incircle

Related topic

Incenter

Related topic

Symmedian

Related topic

TrigonometricVersionOfCevasTheorem

Generated on Fri Feb 9 21:51:24 2018 by

LaTeXML
Gergonne triangle

Gergonne triangle

Let

A

⁢

B

⁢

C

be a

triangle

and

D

,

E

,

F

where the

incircle

touches the sides

B

⁢

C

,

C

⁢

A

,

A

⁢

B

respectively. Then triangle

D

⁢

E

⁢

F

is called the

Gergonne triangle

or

contact triangle

of

A

⁢

B

⁢

C

.

Title

Gergonne triangle

Canonical name

GergonneTriangle

Date of creation

2013-03-22 12:11:14

Last modified on

2013-03-22 12:11:14

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Synonym

contact triangle

Related topic

Triangle

Related topic

GergonnePoint

Related topic

Incircle

Related topic

Incenter

Related topic

Centroid

Related topic

Grafix

Generated on Fri Feb 9 21:51:26 2018 by

LaTeXML
great circle

great circle

The

intersection

of a sphere with a plane that

passes through

the center of the sphere is called a

great circle

. Note that it is

equivalent

to say that a great circle of a sphere is any circle that lies on the surface of the sphere and has maximum

circumference

. Geographically speaking,

longitudes

are examples of great circles; however, with the exception of the equator,

no

latitude

is a great circle.

Infinitely many great circles pass through two

antipodal points

of a sphere. Otherwise, two distinct points on a sphere determine a unique great circle.

An arc of a great circle is called a

great arc

.

Note that great circles and great arcs are

geodesics

of the surface of the sphere on which they lie. Thus, in

spherical geometry

, if a sphere is serving as the model, then are defined to be great circles of the sphere, and are defined to be great arcs of the sphere.

Title

great circle

Canonical name

GreatCircle

Date of creation

2013-03-22 16:06:02

Last modified on

2013-03-22 16:06:02

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Related topic

VolumeOfSphericalCapAndSphericalSector

Defines

great arc

Generated on Fri Feb 9 21:51:28 2018 by

LaTeXML
height

height

Let

A

⁢

B

⁢

C

be a given

triangle

. A height of

A

⁢

B

⁢

C

is a

line segment

drawn from a vertex to the opposite side (or its prolongations) and

perpendicular

to it. So we have three heights in any triangle. The three heights are always

concurrent

and the common point is called the

orthocenter

. In

Euclidean geometry

the
length of the segment ”height” is sometimes referred to as the height.

In the following figure,

A

⁢

D

,

B

⁢

E

and

C

⁢

F

are heights of

A

⁢

B

⁢

C

.

Title

height

Canonical

name

Height

Date of creation

2013-03-22 11:55:39

Last modified on

2013-03-22 11:55:39

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

15

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 51-00

Related topic

Triangle

Related topic

Median

Related topic

Orthocenter

Related topic

Cevian

Related topic

BaseAndHeightOfTriangle

Generated on Fri Feb 9 21:51:30 2018 by

LaTeXML
Heron’s formula

Heron’s formula

The area of a

triangle

with side lengths

a

,

b

,

c

is

△

=

s

⁢

(

s

-

a

)

⁢

(

s

-

b

)

⁢

(

s

-

c

)

where

s

=

a

+

b

+

c

2

(the

semiperimeter

).

Title

Heron’s formula

Canonical name

HeronsFormula

Date of creation

2013-03-22 11:43:54

Last modified on

2013-03-22 11:43:54

Owner

drini (3)

Last modified by

drini (3)

Numerical id

11

Author

drini (3)

Entry type

Theorem

Classification

msc 51-00

Classification

msc 11A51

Classification

msc 55-01

Classification

msc 55-00

Related topic

Triangle

Related topic

BrahmaguptasFormula

Generated on Fri Feb 9 21:51:31 2018 by

LaTeXML
hexagon

hexagon

An

hexagon

is a

6

-sided

polygon

. The most commonly quoted hexagon is a

regular

(

http://

planetmath

.org/RegularPolygon

) hexagon, having

congruent

sides and congruent

interior angles

. Below is an example of a

regular hexagon

:

Below are some

properties

of regular hexagons in

Euclidean geometry

:

•

The measure (

http://planetmath.org/AngleMeasure

) of any interior angle of a regular hexagon is

120

∘

.

•

The smallest

n

for which a regular

n

-gon has

diagonals

which are not congruent is

n

=

6

. For example, in the regular hexagon below, the diagonal drawn in blue and the one drawn in red are not congruent.

•

The side of a regular hexagon has the same length as the radius of the circle circumscribing it. This fact is illustrated below.

From the last remark, it is easy to see that a regular hexagon is

constructible

using compass and straightedge.

Title

hexagon

Canonical

name

Hexagon

Date of creation

2013-03-22 12:10:47

Last modified on

2013-03-22 12:10:47

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

16

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Related topic

Polygon

Related topic

Pentagon

Generated on Fri Feb 9 21:51:33 2018 by

LaTeXML
hypotenuse

hypotenuse

Let

A

⁢

B

⁢

C

a

right triangle

in a

Euclidean geometry

with

right angle

at

C

. Then

A

⁢

B

is called the

hypotenuse

of

A

⁢

B

⁢

C

.

The

midpoint

P

of the hypotenuse coincides with the

circumcenter

of the triangle, so it is equidistant from the three vertices. When the triangle is

inscribed

on his

circumcircle

, the hypotenuse becomes a

diameter

. So the

distance

from

P

to the vertices is precisely the

circumradius

.

The hypotenuse’s length can be calculated by means of the

Pythagorean theorem

:

c

=

a

2

+

b

2

Title

hypotenuse

Canonical

name

Hypotenuse

Date of creation

2013-03-22 12:02:58

Last modified on

2013-03-22 12:02:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51-00

Synonym

hypothenuse

Related topic

Triangle

Related topic

RightTriangle

Related topic

PythagorasTheorem

Related topic

Sohcahtoa

Generated on Fri Feb 9 21:51:35 2018 by

LaTeXML
isogonal conjugate

isogonal conjugate

Let

△

⁢

A

⁢

B

⁢

C

be a

triangle

,

A

⁢

L

the

angle bisector

of

∠

⁢

B

⁢

A

⁢

C

and

A

⁢

X

any line

passing through

A

. The

isogonal conjugate

line to

A

⁢

X

is the line

A

⁢

Y

obtained by reflecting the line

A

⁢

X

on the angle bisector

A

⁢

L

.

In the picture

∠

⁢

Y

⁢

A

⁢

L

=

∠

⁢

L

⁢

A

⁢

X

. This is the reason why

A

⁢

X

and

A

⁢

Y

are called isogonal conjugates, since they form the same angle with

A

⁢

L

. (iso= equal, gonal = angle).

Let

P

be a point on the plane. The lines

A

⁢

P

,

B

⁢

P

,

C

⁢

P

are

concurrent

by construction. Consider now their isogonals conjugates (

reflections

on the inner angle bisectors). The isogonals conjugates will also concurr by the

fundamental theorem on isogonal lines

, and their

intersection

point

Q

is called the isogonal conjugate of

P

.

If

Q

is the isogonal conjugate of

P

, then

P

is the isogonal conjugate of

Q

so both are often referred as an isogonal conjugate pair.

An example of isogonal conjugate pair is found by looking at the

centroid

of the triangle and the

Lemoine point

.

Title

isogonal conjugate

Canonical name

IsogonalConjugate

Date of creation

2013-03-22 13:01:13

Last modified on

2013-03-22 13:01:13

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Related topic

Symmedian

Related topic

LemoinePoint

Related topic

FundamentalTheoremOnIsogonalLines

Defines

isogonal conjugate pair

Defines

isogonal

Generated on Fri Feb 9 21:51:36 2018 by

LaTeXML
isosceles triangle

isosceles triangle

An

isosceles triangle

is a

triangle

with two

congruent

sides. The angles opposite these two sides are the

base angles

and the angle between those sides is the

vertex angle

of the triangle.

This definition implies that any

equilateral triangle

is isosceles
too, but there are isosceles triangles that are not

equilateral

.

In any isosceles triangle, the angles opposite to the congruent sides
are also congruent. In fact, this condition could be used to give an
alternate definition of isosceles, since a triangle is isosceles if
and only if it has two congruent angles.

In an equilateral triangle, the height (

http://planetmath.org/BaseAndHeightOfTriangle

), the median and the

bisector

to
the third side are the same line.

Title

isosceles triangle

Canonical name

IsoscelesTriangle

Date of creation

2013-03-22 11:44:12

Last modified on

2013-03-22 11:44:12

Owner

drini (3)

Last modified by

drini (3)

Numerical id

17

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Classification

msc 49J20

Classification

msc 49J30

Classification

msc 49-01

Synonym

isosceles

Related topic

Triangle

Related topic

RightTriangle

Related topic

EquilateralTriangle

Related topic

EquivalentConditionsForTriangles

Related topic

EquiangularTriangle

Related topic

RegularTriangle

Defines

base angle

Defines

vertex angle

Generated on Fri Feb 9 21:51:40 2018 by

LaTeXML
isosceles triangle theorem

isosceles triangle theorem

The following

theorem

holds in

geometries

in which

isosceles triangle

can be defined and in which SSS, AAS, and SAS are all valid. Specifically, it holds in

Euclidean geometry

and

hyperbolic geometry

(and therefore in

neutral geometry

).

Theorem 1

().

Let

△

⁢

A

⁢

B

⁢

C

be an isosceles triangle such that

A

⁢

B

¯

≅

A

⁢

C

¯

. Let

D

∈

B

⁢

C

¯

. Then the following are

equivalent

:

1.

A

⁢

D

¯

is a median

2.

A

⁢

D

¯

is an

altitude

3.

A

⁢

D

¯

is the

angle bisector

of

∠

⁢

B

⁢

A

⁢

C

Proof.

1

⇒

2

: Since

A

⁢

D

¯

is a median,

B

⁢

D

¯

≅

C

⁢

D

¯

. Since we have

•

A

⁢

B

¯

≅

A

⁢

C

¯

•

B

⁢

D

¯

≅

C

⁢

D

¯

•

A

⁢

D

¯

≅

A

⁢

D

¯

by the reflexive property (

http://planetmath.org/

Reflexive

) of

≅

we can use SSS to conclude that

△

⁢

A

⁢

B

⁢

D

≅

△

⁢

A

⁢

C

⁢

D

. By CPCTC,

∠

⁢

A

⁢

D

⁢

B

≅

∠

⁢

A

⁢

D

⁢

C

. Thus,

∠

⁢

A

⁢

D

⁢

B

and

∠

⁢

A

⁢

D

⁢

C

are

supplementary

(

http://planetmath.org/SupplementaryAngle

)

congruent

angles. Hence,

A

⁢

D

¯

and

B

⁢

C

¯

are

perpendicular

. It follows that

A

⁢

D

¯

is an altitude.

2

⇒

3

: Since

A

⁢

D

¯

is an altitude,

A

⁢

D

¯

and

B

⁢

C

¯

are perpendicular. Thus,

∠

⁢

A

⁢

D

⁢

B

and

∠

⁢

A

⁢

D

⁢

C

are

right angles

and therefore congruent. Since we have

•

∠

⁢

B

≅

∠

⁢

C

by the theorem on angles of an isosceles triangle

•

∠

⁢

A

⁢

D

⁢

B

≅

∠

⁢

A

⁢

D

⁢

C

•

A

⁢

D

¯

≅

A

⁢

D

¯

by the reflexive property of

≅

we can use AAS to conclude that

△

⁢

A

⁢

B

⁢

D

≅

△

⁢

A

⁢

C

⁢

D

. By CPCTC,

∠

⁢

B

⁢

A

⁢

D

≅

∠

⁢

C

⁢

A

⁢

D

. It follows that

A

⁢

C

¯

is the angle bisector of

∠

⁢

B

⁢

A

⁢

C

.

3

⇒

1

: Since

A

⁢

D

¯

is an angle bisector,

∠

⁢

B

⁢

A

⁢

D

≅

∠

⁢

C

⁢

A

⁢

D

. Since we have

•

A

⁢

B

¯

≅

A

⁢

C

¯

•

∠

⁢

B

⁢

A

⁢

D

≅

∠

⁢

C

⁢

A

⁢

D

•

A

⁢

D

¯

≅

A

⁢

D

¯

by the reflexive property of

≅

we can use SAS to conclude that

△

⁢

A

⁢

B

⁢

D

≅

△

⁢

A

⁢

C

⁢

D

. By CPCTC,

B

⁢

D

¯

≅

C

⁢

D

¯

. It follows that

A

⁢

D

¯

is a median.
∎

Remark

: Another equivalent (

http://planetmath.org/Equivalent3

) condition for

A

⁢

D

¯

is that it is the

perpendicular bisector

of

B

⁢

C

¯

; however, this fact is usually not included in the statement of the

Isosceles Triangle Theorem

.

Title

isosceles triangle theorem

Canonical name

IsoscelesTriangleTheorem

Date of creation

2013-03-22 17:12:12

Last modified on

2013-03-22 17:12:12

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 51-00

Classification

msc 51M04

Related topic

ConverseOfIsoscelesTriangleTheorem

Generated on Fri Feb 9 21:51:43 2018 by

LaTeXML
kite

kite

A

kite

or

deltoid

is a

quadrilateral

with two pairs of equal sides, each pair consisting of

adjacent sides

. Contrast with

parallelograms

, where the equal sides are opposite.

The pairs of equal sides imply several properties:

•

One

diagonal

divides the kite into two

isosceles triangles

, and the other divides the kite into two

congruent

triangles

.

•

The angles between the sides of unequal length are equal. In the picture, they are both equal to the sum of the blue angle with the red angle.

•

The diagonals are

perpendicular

.

•

If

d

1

and

d

2

are the lengths of the diagonals, then the area is

A

=

d

1

⁢

d

2

2

Alternatively, if

a

and

b

are the lengths of the sides, and

θ

the angle between unequal sides, then the area is

A

=

a

⁢

b

⁢

sin

⁡

θ

.

•

A kite possesses an inscribed circle. That is, there exists a circle that is

tangent

(touches) the four sides.

•

Kites always possess at least one

symmetry

axis, being the diagonal that divides it into two congruent triangle.

When all the side lengths are the same, the kite becomes a

rhombus

, and when both diagonals have the same length, the kite becomes a square.

Title

kite

Canonical name

Kite

Date of creation

2013-03-22 15:49:22

Last modified on

2013-03-22 15:49:22

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 51-00

Synonym

deltoid

Related topic

Parallelogram

Related topic

Quadrilateral

Related topic

Rhombus

Generated on Fri Feb 9 21:51:45 2018 by

LaTeXML
legs

legs

The

legs

of a

right triangle

are the two sides which are not the

hypotenuse

.

Above: Various right triangles, with legs in red.

Note that there are no legs for

scalene

or non-right

isosceles triangles

, just as there is no notion of hypotenuse for these special triangles. An exception to this is that the two equal sides of an isosceles triangle may occasionally be referred to as “legs”, but this is uncommon.

Title

legs

Canonical

name

Legs

Date of creation

2013-03-22 12:06:02

Last modified on

2013-03-22 12:06:02

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 51-00

Generated on Fri Feb 9 21:51:46 2018 by

LaTeXML
Lemoine circle

Lemoine circle

If through the

Lemoine point

of a

triangle

are drawn

parallels

to the sides, the six points where these intersect the circle lie all on a same circle. This circle is called the

Lemoine circle

of the triangle

Title

Lemoine circle

Canonical name

LemoineCircle

Date of creation

2013-03-22 12:11:05

Last modified on

2013-03-22 12:11:05

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Related topic

Triangle

Related topic

Symmedian

Related topic

LemoinePoint

Related topic

Incircle

Generated on Fri Feb 9 21:51:48 2018 by

LaTeXML
Lemoine point

Lemoine point

The

Lemoine point

of a

triangle

, is the

intersection

point of its three

symmedians

. (That is, the

isogonal conjugate

of the

centroid

).

It is related with the

Gergonne point

by the following result:

On any triangle

A

⁢

B

⁢

C

, the Lemoine point of its

Gergonne triangle

is the Gergonne point of

A

⁢

B

⁢

C

.

In the picture, the blue lines are the medians, intersecting an the centroid

G

.
The green lines are anglee

bisectors

intersecting at the

incentre

I

and the red lines are symmedians. The symmedians intersect at Lemoine point

L

.

Title

Lemoine point

Canonical name

LemoinePoint

Date of creation

2013-03-22 12:11:02

Last modified on

2013-03-22 12:11:02

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

Related topic

Triangle

Related topic

Symmedian

Related topic

LemoineCircle

Related topic

Incircle

Related topic

Centroid

Related topic

Incenter

Related topic

GergonnePoint

Related topic

Isogonal

Related topic

IsogonalConjugate

Related topic

FundamentalTheoremOnIsogonalLines

Generated on Fri Feb 9 21:51:50 2018 by

LaTeXML
linear pair

linear pair

Two angles are a

linear pair

if the angles are

adjacent

and the two unshared rays form a line. Below is an example of a linear pair:

The

linear pair postulate

states that two angles that form a linear pair are

supplementary

(

http://planetmath.org/Supplementary2

).

Title

linear pair

Canonical

name

LinearPair

Date of creation

2013-03-22 17:35:38

Last modified on

2013-03-22 17:35:38

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51-00

Defines

linear pair postulate

Generated on Fri Feb 9 21:51:52 2018 by

LaTeXML
magic circles

magic circles

Given

n

circles which

intersect

k

times, the set of circles is referred to as

magic circles

if there exists a numbering (using

{

1

,

…

,

k

}

) of the

intersections

of the

n

circles such that the sum of the numbers at the intersections along each circle is the same

constant

for all circles.

Further reference:
http://www.cgl.uwaterloo.ca/ smann/Magic/

Title

magic circles

Canonical

name

MagicCircles

Date of creation

2013-03-22 14:15:22

Last modified on

2013-03-22 14:15:22

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 51-00

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

LaTeXML
medial triangle

medial triangle

The

medial triangle

of a

triangle

△

⁢

A

⁢

B

⁢

C

is the triangle formed by joining the

midpoints

of the sides of the triangle

△

⁢

A

⁢

B

⁢

C

.

Here,

△

⁢

A

′

⁢

B

′

⁢

C

′

is the medial triangle.
The

incircle

of the medial triangle is called the

Spieker circle

and the

incenter

is called the

Spieker center

.
The

circumcircle

of the medial triangle is called the

medial circle

.

An important

property

of the medial triangle is that the medial triangle

△

⁢

A

′

⁢

B

′

⁢

C

′

of the medial triangle

△

⁢

D

⁢

E

⁢

F

of

△

⁢

A

⁢

B

⁢

C

is

similar

to

△

⁢

A

⁢

B

⁢

C

.

Title

medial triangle

Canonical

name

MedialTriangle

Date of creation

2013-03-22 13:11:17

Last modified on

2013-03-22 13:11:17

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Definition

Classification

msc 51-00

Synonym

auxiliary triangle

Defines

Spieker center

Defines

Spieker circle

Defines

medial circle

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

LaTeXML