Παρασκευή, 2 Ιουλίου 2010

A SET ADDITION AND CONCURRENT CIRCLES

Let W = {a1,a2,a3,a4,....,an} be a finite set and S = {A1,A2,...,Ak} a set of subsets of W. We define the addition operation + in S:

X + Y = (X - (X ∩ Y)) U (W - (X U Y))

Example:

W = {1,2,3,4,5}, S = {{1,2,3},{1,3,4},{2,3,5}}

{1,2,3} + {1,3,4} = {2} U {5} = {2,5}

Examples from Geometry of closed sets S under the addition +
(ie if X, Y belong in S, then X + Y belongs in S as well).

FOUR CONCURRENT CIRCLES.

Let 123 be a triangle, 4 a point, 1', 2' and 3' the circles (234), (341) and (412), resp. and 4' an arbitrary circle passing through 4.

Denote:

5 := 4' ∩ 1' - {4}

6 := 4' ∩ 2' - {4}

7 := 4' ∩ 3' - {4}

(ie the other than point 4 intersections of the circle 4' with the circles 1',2',3', resp.)

5' := the circle (167), 6' := the circle (275), 7' := the circle (356)

Theorem:

The circles 5',6',7' concur at a point 8 on the circle
(123) := 8'

The circle 4' may be of infinite radius (ie be a line):

Consider the sets:

W = {1,2,3,4,5,6,7,8}, S = {1',2',3',4',6',7',8'}

where 1', 2',...,8' are sets of four concyclic points:
(not be confused with the circles 1', 2',... above:
1' above is the circle passing through the four points 2,3,4,5, while 1' is now the set containing the four points 2,3,4,5)

1' = {2,3,4,5}
2' = {1,3,4,6}
3' = {1,2,4,7}
4' = {4,5,6,7}
5' = {1,6,7,8}
6' = {2,5,7,8}
7' = {3,5,6,8}
8' = {1,2,3,8}

We have:

1' + 1' = {1,6,7,8} = 5'
1' + 2' = {2,5,7,8} = 6'
1' + 3' = {3,5,6,8} = 7'
1' + 4' = {1,2,3,8} = 8'
1' + 5' = {2,3,4,5} = 1'
1' + 6' = {1,3,4,6} = 2'
1' + 7' = {1,2,4,7} = 3'
1' + 8' = {4,5,6,7} = 4'

etc

+ | 1' 2' 3' 4' 5' 6' 7' 8'
------------------------------------
1' | 5' 6' 7' 8' 1' 2' 3' 4'

2' | 5' 6' 7' 8' 1' 2' 3' 4'

3' | 5' 6' 7' 8' 1' 2' 3' 4'

4' | 5' 6' 7' 8' 1' 2' 3' 4'

5' | 5' 6' 7' 8' 1' 2' 3' 4'

6' | 5' 6' 7' 8' 1' 2' 3' 4'

7' | 5' 6' 7' 8' 1' 2' 3' 4'

8' | 5' 6' 7' 8' 1' 2' 3' 4'

"Dual":

Denote:

W = {1',2',3',4',5',6',7',8'}, S = {1,2,3,4,5,6,7,8}

where: 1,2,..., are sets of four circles passing through a point
(not be confused with the points 1, 2,... above:
1 above is the common point of the circles 2',3',5',8', while 1 is now the set of the four circles passing through the point 1):

1 = {2',3',5',8'}
2 = {1',3',6',8'}
3 = {1',2',7',8'}
4 = {1',2',3',4'}
5 = {1',4',6',7'}
6 = {2',4',5',7'}
7 = {3',4',5',6'}
8 = {5',6',7',8'}

We have:

1 + 1 = {1',4',6',7'} = 5
1 + 2 = {2',4',5',7'} = 6
1 + 3 = {3',4',5',6'} = 7
1 + 4 = {5',6',7',8'} = 8
1 + 5 = {2',3',5',8'} = 1
1 + 6 = {1',3',6',8'} = 2
1 + 7 = {1',2',7',8'} = 3
1 + 8 = {1',2',3',4'} = 4

etc

+ | 1 2 3 4 5 6 7 8
-------------------------------
1 | 5 6 7 8 1 2 3 4

2 | 5 6 7 8 1 2 3 4

3 | 5 6 7 8 1 2 3 4

4 | 5 6 7 8 1 2 3 4

5 | 5 6 7 8 1 2 3 4

6 | 5 6 7 8 1 2 3 4

7 | 5 6 7 8 1 2 3 4

8 | 5 6 7 8 1 2 3 4

NOTES:

1. References of the configurations:
Antreas P. Hatzipolakis (et al): Hyacinthos Thread Nice!
Antreas P. Hatzipolakis (et al): Hyacinthos Thread 3+1 Circles

2. Generalization