**Introduction to Polygon Geometry**

This encyclopedia is giving illustrations and a description of properties related to Poly-Figures.

Here a Poly-Figure is defined as a plane geometrical figure of n points and/or n lines, where n is a natural number > 1.

In this Encyclopedia we will use these terms:

•

**n-Gon**, meaning a figure consisting of n points and n lines connected in a fixed order,

•

**n-Point**, meaning a figure consisting of n random points without order,

•

**n-Line**, meaning a figure consisting of n random lines without order,

where n is a natural number >1.

For the different values of n we have these Poly-Figures:

Well-known figure consisting of 3 points and 3 lines.

* Triangle = 3-Point = 3-Line = 3-Gon.

Described in the Encyclopedia of Triangle Centers ETC. See Ref-12.

* Quadrangle (4-Point),

* Quadrilateral (4-Line),

* Quadrigon (4-Gon).

Described in the Encyclopedia of Quadri Figures EQF.

* Pentangle (5-Point)

* Pentalateral (5-Line)

* Pentagon (5-Gon)

n = 6 Hexagon

n = 7 Heptagon

n = 8 Octagon

n = 9 Nonagon or Enneagon

n = 10 Decagon

n = 11 Hendecagon

n = 12 Dodecagon

n =13 Tridecagon

n = 14 Tetradecagon

n = 15 Pentadecagon

n = 16 Hexadecagon

n = 17 Heptadecagon

n = 18 Octadecagon

n = 19 Enneadecagon

n = 20 Icosagon

n = 30 Triacontagon

n = 100 Hectogon

etc.

A more extensive list can be found at Ref-65.

In EPF three basic platforms occur:

• n-Line, meaning a figure consisting of n random lines without order.

All objects of an n-Line will be prefixed with “nL-“.

• n-Point, meaning a figure consisting of n random points without order.

All objects of an n-Point will be prefixed with “nP-“.

• n-Gon, meaning a figure consisting of n points and n lines cyclically connected in a fixed order. All objects of an n-Gon will be prefixed with “nG-“.

*n = 3 Triangle*Well-known figure consisting of 3 points and 3 lines.

* Triangle = 3-Point = 3-Line = 3-Gon.

Described in the Encyclopedia of Triangle Centers ETC. See Ref-12.

*n = 4 Quadri-Figure** Quadrangle (4-Point),

* Quadrilateral (4-Line),

* Quadrigon (4-Gon).

Described in the Encyclopedia of Quadri Figures EQF.

*n = 5 Penta-Figure** Pentangle (5-Point)

* Pentalateral (5-Line)

* Pentagon (5-Gon)

*for Polygons or n-Gons when n>5 are:***Familiar names**n = 6 Hexagon

n = 7 Heptagon

n = 8 Octagon

n = 9 Nonagon or Enneagon

n = 10 Decagon

n = 11 Hendecagon

n = 12 Dodecagon

n =13 Tridecagon

n = 14 Tetradecagon

n = 15 Pentadecagon

n = 16 Hexadecagon

n = 17 Heptadecagon

n = 18 Octadecagon

n = 19 Enneadecagon

n = 20 Icosagon

n = 30 Triacontagon

n = 100 Hectogon

etc.

A more extensive list can be found at Ref-65.

**n-Lines, n-Points and n-Gons**In EPF three basic platforms occur:

• n-Line, meaning a figure consisting of n random lines without order.

All objects of an n-Line will be prefixed with “nL-“.

• n-Point, meaning a figure consisting of n random points without order.

All objects of an n-Point will be prefixed with “nP-“.

• n-Gon, meaning a figure consisting of n points and n lines cyclically connected in a fixed order. All objects of an n-Gon will be prefixed with “nG-“.

Recursive processesRecursive processes

Like described before a Polygon has n variable points/lines whether or not in a fixed order.

There are similar points in Polygons for all n > 2.

For example when n = 3 we have a 3-Line or a triangle and in a triangle we have a circumcircle and a circumcircle has a circumcenter X(3).

When n = 4 we have a 4-Line or a quadrilateral containing 4 component 3-Lines/Triangles, each having a circumcenter X(3). These 4 circumcenters lie on a circle called “4L-Centercircle”. This “4L-Centercircle” has a circumcenter QL-P4.

When n = 5 we have a 5-Line or a Pentalateral containing 5 component 4-Lines/Quadrilaterals, each having a 4-Line circumcenter. These 5 circumcenters lie on a circle called “5L-Centercircle”. This “5L-Centercircle” has a 5L-circumcenter.

Etc.

This takes a lot of words just describing a relatively simple recursive construction method.

By using simple notation techniques we can simplify the wording of these kind of statements.

For example above sentences can be simplified by stating:

“An n-Line contains n (n-1)-Lines. So in an n-Line n (n-1)L-circumcenters can be constructed. These circumcenters are always concyclic and therefore define an nL-center-circle having an nL-circumcenter. This recursive process starts with the circumcircle of a 3-Line.”

There are similar points in Polygons for all n > 2.

For example when n = 3 we have a 3-Line or a triangle and in a triangle we have a circumcircle and a circumcircle has a circumcenter X(3).

When n = 4 we have a 4-Line or a quadrilateral containing 4 component 3-Lines/Triangles, each having a circumcenter X(3). These 4 circumcenters lie on a circle called “4L-Centercircle”. This “4L-Centercircle” has a circumcenter QL-P4.

When n = 5 we have a 5-Line or a Pentalateral containing 5 component 4-Lines/Quadrilaterals, each having a 4-Line circumcenter. These 5 circumcenters lie on a circle called “5L-Centercircle”. This “5L-Centercircle” has a 5L-circumcenter.

Etc.

This takes a lot of words just describing a relatively simple recursive construction method.

By using simple notation techniques we can simplify the wording of these kind of statements.

For example above sentences can be simplified by stating:

“An n-Line contains n (n-1)-Lines. So in an n-Line n (n-1)L-circumcenters can be constructed. These circumcenters are always concyclic and therefore define an nL-center-circle having an nL-circumcenter. This recursive process starts with the circumcircle of a 3-Line.”

**m-Lines and p-Lines**By introducing another notation we can even simplify these wordings.

In the recursive process we often deal with (n-1)-Lines or (n+1)-Lines.

To indicate that we talk about an n-Line of a lower or upper level we can talk about m-Lines or p-Lines, where “m” and “p” resp. are meaning ”minus 1” and “plus 1”.

So now we can say:

“An n-Line contains n m-Lines. So in an n-Line n mL-circumcenters can be constructed.”

instead of:

“An n-Line contains n (n-1)-Lines. So in an n-Line n (n-1)L-circumcenters can be constructed.

This notation will be used whenever convenient.

**The neos-system: n-Points, e-Points, o-Points and s-Points**In the Encyclopedia of Polygon Geometry different types of points will be distinguished.

an n-Point is a recursive point that occurs in all n-Lines for n=natural number >2.

an e-Point is a recursive point that occurs in all n-Lines for n=even number >2.

an o-Point is a recursive point that occurs in all n-Lines for n=odd number >2.

an s-Point is a non-recursive but specific point only occurring in an n-Line for n = specific natural number > 2.

For example the notation for a point will be:

• nL-n-P1, meaning general-recursive Point 1 in an n-Line, where n can be 3,4,5,6,….

• nL-e-P1, meaning even-recursive Point 1 in an n-Line, where n can be even numbers 4,6,8,10,…

• nL-o-P1, meaning odd-recursive Point 1 in an n-Line, where n can be odd numbers 3,5,7,9,….

• nL-s-P1, meaning specific non-recursive Point 1 in an n-Line, where n is specifically 3,4,5,6,….

• Etc.

This implies that we will have different sets of points.

n-Points, e-Points, o-Points will be described in general.

s-Points will be described for the fixed number of n it occurs with.

The same infixes -n-, -e-, -o-, -s- will also be used for Lines, Circles, Cubics, Quartics, Transformations, etc.

Here are some examples for n-Lines:

5L-n-P1, meaning general-recursive Point 1 in a 5-Line

6L-e-P1, meaning even-recursive Point 1 in a 6-Line

7L-o-P1, meaning odd-recursive Point 1 in a 7-Line

8L-s-P1, meaning specific non-recursive Point 1 in an 8-Line

5L-n-P1, meaning general-recursive Point 1 in a 5-Line

6L-e-P1, meaning even-recursive Point 1 in a 6-Line

7L-o-P1, meaning odd-recursive Point 1 in a 7-Line

8L-s-P1, meaning specific non-recursive Point 1 in an 8-Line

**Central Points/Centers**Described points in the Encyclopedia of Polygon Geometry actually will be “central points” or “centers”. The meaning of a central point/center best can be given with an example.

For example in 3-Line/triangle we have 3 lines L1,L2,L3.

We have the intersection point S12 of L1 and L2. This not a central point.

We have the circumcenter O of the circumcircle. This is a central point.

S12 is not a central point in a 3-Line because it is not equally dependent on the 3 basic elements of a 3-Line, namely L1,L2,L3.

However O is a central point in a 3-Line because it is equally dependent on the 3 basic elements of a 3-Line, namely L1,L2,L3.

The same can be done in a 4-Line figure, etc.

In literature little is written about central points in a Polygon.

Clark Kimberling defines in ETC (see Ref-12) a triangle center like this:

Suppose a point P has a trilinear representation

f(A,B,C) : g(A,B,C) : h(A,B,C) such that

(i) g(A,B,C ) = f(B,C,A)

and h(A,B,C) = f(C,A,B);

(ii) f(A,C,B) = f(A,B,C);

(iii) if P is written as

u(a,b,c) : u(b,c,a) : u(c,a,b), where a,b,c are the side lengths of triangle ABC, then u is homogeneous in the variables a,b,c. (By the law of sines and (i), such u must exist.)

Then P is a triangle center, or simply a center.

Benedetto Scimemi proposes in his document “Central Points of the Complete Quadrangle” (see Ref-36):

Let E be the Euclidean plane; a (ngonal)central point P is a symmetric mapping: En _→ E which commutes with all similarities φ (in the sense that P(φ(Ai))=φ(P(Ai))). Likewise one defines central lines, central scalars, central conics etc. If this definition is studied analytically, some interesting algebraic questions naturally arise.

Important for the Encyclopedia of Polygon Geometry is that we only describe Central Points/Centers and related Central Lines, Central Conics, etc.

**How many Poly-Points/Centers do potentially exist?**When using the notion of Point here we actually mean a Central Point or Center. See paragraph just before.

In

**very many Points are described in Ref-12, the Encyclopedia of Triangle Centers (ETC). And that’s only just the beginning.**

*Triangle Geometry*Points can be combined giving rise to other points. So it looks like there is no end in the number of Triangle points.

In

**less points are described.**

*Quadrilateral Geometry*However there is the

**DT-method**(Diagonal Triangle-method) for making 4L-points/4P-points from ETC-points (being 3L-points/3P-points):

For Quadrilaterals (4-Line figure) as well as Quadrangles (4-Point figure) we have a Diagonal Triangle (resp. QL-Tr1 and QA-Tr1). These are Triangles and every ETC-point in these triangles become a Quadrilateral-/ Quadrangle-Point in the system of the Reference Quadrilateral-/ Quadrangle because their construction is strictly symmetric. Maybe they are not very interesting points because generally there are hardly relations with existing Quadrilateral-/ Quadrangle-Points. However the principle counts.

Then there is also the

**Ref/Per2-method**of making 4L-points from ETC-points (3L-points):

Let Ref be a Reference Quadrilateral of lines L-1,L-2,L-3,L-4.

Let P-i = ETC-point Px of triangle (Lj.Lk.Ll), where (i,j,k,l) are different numbers from (1,2,3,4).

Let Lp-i be the perpendiculars from P-i on L-i (i=1,2,3,4).

Now we have a 1st generation perpendicular quadrilateral Per1.

By doing the same construction on Per1 (instead of on Ref ) we get a 2nd generation perpendicular quadrilateral Per2.

For several ETC-points it has been checked that all the time Ref is homothetic with Per2 (except for cases with extremities). Because the construction is strictly symmetric there will be for all these ETC-points Px a QL-homothetic center (HC) QL-Px.

See QFG#1937,#1938.

Let P-i = ETC-point Px of triangle (Lj.Lk.Ll), where (i,j,k,l) are different numbers from (1,2,3,4).

Let Lp-i be the perpendiculars from P-i on L-i (i=1,2,3,4).

Now we have a 1st generation perpendicular quadrilateral Per1.

By doing the same construction on Per1 (instead of on Ref ) we get a 2nd generation perpendicular quadrilateral Per2.

For several ETC-points it has been checked that all the time Ref is homothetic with Per2 (except for cases with extremities). Because the construction is strictly symmetric there will be for all these ETC-points Px a QL-homothetic center (HC) QL-Px.

See QFG#1937,#1938.

This is not only true for Quadri Geometry but also for Polygon Geometry.

There is also at least the

**MVP-method**for making nL-points and nP-points from ETC-points (3L-points):

Every Triangle Center can be transferred to a corresponding point in an n-Line by a simple recursive construction. The resulting point which will be called an nL-MVP Center, where MVP is the abbreviation for Mean Vector Point.

Definition: A Mean Vector Point (MVP) is the mean of a bunch of vectors with identical origin. It is constructed by adding these vectors and then dividing the resultant vector by n. The Mean Vector Point is the endpoint of the divided resultant vector. In all n-Lines we can use any random point as origin. The endpoint of the resultant vector will be invariant for all different origins.

When X(i) is a triangle Center we define the nL-MVP X(i)-Center as the Mean Vector Point of the n (n-1)L-MVP X(i)-Centers.

When the (n-1)L-MVP X(i)-Centers aren’t known they can be constructed from the MVP X(i)-Centers another level lower, according to the same definition. By applying this definition to an increasingly lower level finally the level is reached of the 3L-MVP X(i)-Center, which simply is the X(i) Triangle Center. Then it can be “rolled up” to the starting level.

See QFG#869,#873,#878,#881.

Definition: A Mean Vector Point (MVP) is the mean of a bunch of vectors with identical origin. It is constructed by adding these vectors and then dividing the resultant vector by n. The Mean Vector Point is the endpoint of the divided resultant vector. In all n-Lines we can use any random point as origin. The endpoint of the resultant vector will be invariant for all different origins.

When X(i) is a triangle Center we define the nL-MVP X(i)-Center as the Mean Vector Point of the n (n-1)L-MVP X(i)-Centers.

When the (n-1)L-MVP X(i)-Centers aren’t known they can be constructed from the MVP X(i)-Centers another level lower, according to the same definition. By applying this definition to an increasingly lower level finally the level is reached of the 3L-MVP X(i)-Center, which simply is the X(i) Triangle Center. Then it can be “rolled up” to the starting level.

See QFG#869,#873,#878,#881.

Still it is too early to say that there are more Poly-Figure-points then Triangle-points because possibly there are general mechanisms creating ETC-points from nL- or nP-points.

**Documentation**Everything in EPG is documented with references when known.

When no references are mentioned the information is found and described by the author of EPG. Even then it is possible that items and properties were known before. When you know this kind of information or when you have other remarks or questions please let me know by mail.

Several items are proven. When known there will be a reference to the corresponding article. However especially with Poly-figures it often is very hard to give a full proof for the validity of involved item.

Therefore when things are very likely and “proven” with drawing software or algebraic software they still will be mentioned in EPG, often with reference to discussions and waiting for a person who delivers the full proof.