4. n-Lines, n-Points and n-Gons

Within EPG, three fundamental platforms are defined:

• n-Line: a figure consisting of n unordered lines.
  All objects of an n-Line are prefixed with nL-.
• n-Point: a figure consisting of n unordered points.
  All objects of an n-Point are prefixed with nP-.
• n-Gon: a figure consisting of n points and n lines cyclically connected in a fixed order.
  All objects of an n-Gon are prefixed with nG-.

5. Recursive Processes

As previously described, a polygon consists of n variable points and lines, either ordered or unordered. For all n > 2, similar recursive constructions apply.

For example:

• n = 3: A 3-Line or triangle has a circumcircle, which has a circumcenter X(3).
• n = 4: A 4-Line or quadrilateral contains 4 component 3-Lines (triangles), each with a circumcenter X(3). These 4 circumcenters lie on a circle called the 4L-Centercircle, which has a circumcenter QL-P4.
• n = 5: A 5-Line or pentalateral contains 5 component 4-Lines (quadrilaterals), each with a 4-Line circumcenter. These 5 circumcenters lie on a circle called the 5L-Centercircle, which has a circumcenter 5L-Circumcenter.

This recursive construction method, though conceptually simple, often requires extensive wording. By introducing concise notation, we can express these ideas more efficiently.

Example of simplified phrasing:

“An n-Line contains n (n−1)-Lines. Thus, in an n-Line, n (n−1)L-circumcenters can be constructed. These circumcenters are always concyclic and define an nL-Centercircle with an nL-Circumcenter. This recursive process begins with the circumcircle of a 3-Line.”

6. m-Lines and p-Lines

To further streamline notation, we introduce the terms m-Line and p-Line, representing an n-Line of one level lower or higher, respectively:

• m = minus 1 → (n−1)-Line
• p = plus 1 → (n+1)-Line

This allows us to say:

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

instead of the more verbose:

“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 wherever it enhances clarity and brevity.

7. The Neos-System: n-Points, e-Points, o-Points and s-Points

Within the Encyclopedia of Polygon Figures, several types of points are distinguished:

• n-Point: a recursive point occurring in all n-Lines for n a natural number > 2.
• e-Point: a recursive point occurring in all n-Lines for n an even number > 2.
• o-Point: a recursive point occurring in all n-Lines for n an odd number > 2.
• s-Point: a non-recursive but specific point occurring only in an n-Line for a fixed n > 2.

Point notation examples:

• nL-n-P1: general-recursive Point 1 in an n-Line, where n = 3, 4, 5, 6, …
• nL-e-P1: even-recursive Point 1 in an n-Line, where n = 4, 6, 8, 10, …
• nL-o-P1: odd-recursive Point 1 in an n-Line, where n = 3, 5, 7, 9, …
• nL-s-P1: specific non-recursive Point 1 in an n-Line, where n is fixed number 3, 4, 5, 6, …

This implies the existence of distinct point sets:

• n-Points, e-Points, o-Points: described in general terms
• s-Points: described specifically for each fixed value of n

The same infixes -n-, -e-, -o-, -s- are also used for Lines, Circles, Cubics, Quartics, Transformations, and other geometric entities.

8. Literature on Central Points

In existing literature, central points in polygons are rarely discussed. Two notable contributions are presented below, with their original formulations preserved:

Clark Kimberling (ETC, see [12]) defines a triangle center as follows:

Benedetto Scimemi (“Central Points of the Complete Quadrangle”, see [36]) proposes:

For the purposes of the Encyclopedia of Poly Geometry, we restrict our scope to central points (centers) and their related central lines, central conics, and similar constructs.

9. How Many Poly-Points/Centers Potentially Exist?

When using the notion of Point here, we actually mean a Central Point or Center. See the paragraph just before.

Triangle Geometry

In triangle geometry, a vast number of points are described in [12], the Encyclopedia of Triangle Centers (ETC). And that’s only the beginning. Points can be combined, giving rise to other points. It appears there is no end to the number of triangle points.

Quadri Geometry

In quadri geometry, fewer points are described. However, several methods exist to generate new points from ETC-points:

DT-method (Diagonal Triangle Method)

For Quadrilaterals (4-Line figures) and Quadrangles (4-Point figures), we define a Diagonal Triangle (QL-Tr1 and QA-Tr1 respectively). These are triangles, and every ETC-point in these triangles becomes a Quadrilateral-/Quadrangle-Point in the system of the Reference Quadrilateral-/Quadrangle because their construction is strictly symmetric. These points may not be very interesting individually, as they often lack strong relations with existing Quadrilateral-/Quadrangle-Points. However, the principle remains valid.

Ref/Per2-method

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 distinct numbers from (1, 2, 3, 4).
Let Lp-i be the perpendiculars from P-i to L-i (i = 1, 2, 3, 4).
This yields a first-generation perpendicular quadrilateral Per1.
Repeating the construction on Per1 yields a second-generation perpendicular quadrilateral Per2.
For several ETC-points, it has been verified that Ref is homothetic with Per2 (except in extreme cases). Due to the symmetric construction, each ETC-point Px yields a QL-homothetic center (HC) QL-Px.
See QFG#1937, #1938.

Poly Geometry

The same principles apply beyond Quadri Geometry, into general Poly Geometry.

MVP-method (Mean Vector Point)

Every triangle center can be transferred to a corresponding point in an n-Line via a recursive construction. The resulting point is called an nL-MVP Center, where MVP stands for Mean Vector Point.

Definition: A Mean Vector Point (MVP) is the mean of a set of vectors with identical origin. It is constructed by summing the vectors and dividing the resultant vector by n. The MVP is the endpoint of this divided vector. In all n-Lines, any random point can serve as origin. The endpoint of the resultant vector remains invariant across different origins.

When X(i) is a triangle center, the nL-MVP X(i)-Center is defined as the MVP of the n (n−1)L-MVP X(i)-Centers. If the (n−1)L-MVP X(i)-Centers are unknown, they can be constructed from MVP X(i)-Centers one level lower, using the same definition. By recursively applying this process, we eventually reach the 3L-MVP X(i)-Center, which is simply the original triangle center X(i). From there, the construction can be “rolled up” to the desired level.
See QFG#869, #873, #878, #881.

Conclusion

It is still too early to conclude that there are more Poly-Figure points than Triangle points. There may even exist general mechanisms that generate ETC-points from nL- or nP-points.