From Euclid to Combinatorics

[Antreas P. Hatzipolakis]

AN OLD THEOREM OF EUCLIDEAN GEOMETRY:

Let ABC be a triangle.

1,2,3 := three parallel lines through A,B,C, resp.

1',2',3' := three perpendiculars to 1,2,3, through A,B,C, resp.

Denote: i/\ j' = (ij') := (j'i) [so we have A = (11'), B = (22'), C = (33')]

The lines (23')(32ã), (31')(13'), (21ã)(12'), are concurrent
at a point on the Nine Point Circle of the triangle (11')(22')(33') = ABC


      2'     1'      3'
      |      |       |
2 ---B-----21'-----23'---------
      |      |       |
1 --12'-----A------13'---------
      |      |       |
      |      |       |
3 --32'----31'------C----------
      |      |       |


PROBLEMS

Let 1,2,3,...., n be n horizontal lines, and 1',2',3',..., n'
n vertical lines.

Let i/\ j' = (ij') := (j'i)

Two points are "collinear" if they lie on the same horizontal
or vertical line.

Problem #1:
Which is the number f(n) of the  non-ordered triangles whose
the vertices are not "collinear" ?

If n = 3, there are 6 triangles, namely:

(11')(22')(33')
(11')(23')(31')
(12')(23')(31')
(12')(21')(33')
(13')(22')(31')
(13')(21')(32')


Geometry Problem: Where are lying the respective Nine Point
Circles points?


Problem #2:

Let n = 2k + 1

Which is the number c(n) of the  non-ordered triangles with
non "collinear" vertices, that have as vertex the central point
([k+1][k+1]') ?

If k = 1, there are two triangles with vertex the point (22'):

(11')(22')(33')
(13')(22')(31')


Antreas



--