## Τρίτη, 20 Ιουλίου 2010

### THREE CONCURRENT CIRCLES

5. Let 123 be a triangle, 4 a point inside 123 and (1') the circle touching the circles (134), (124) externally and the circle (234) internally at 5, (2') the circle touching the circles (214),(234) externally and the circle (314) internally at 6 and (3') the circle touching the circles (324),(314) externally and the circle (124) internally at 7.

The circles (167),(275),(356) concur at a point 8.

Variation:

Let (0) be the circle touching internally the circles (234), (314), (124) at 5,6,7 resp.

The circles (167),(275),(356) concur at a point 8.

Note:

If 4 is not inside triangle 123, but in the negative side of 23 (ie the side not containing 1), then (1') is the circle touching (314),(124) internally and (234) externally at 5.

## Σάββατο, 3 Ιουλίου 2010

### THREE CONCURRENT CIRCLES

Let 123 be a trianle, 4 a point, and 5,6,7 three points on the circles (423), (431),(412) resp. other than point 4.

Theorem:

The circles (167), (275) and (356) concur at a point (say) 8.

Special Cases:

1. Let the points 4,5,6,7 be concyclic (or collinear). See previous post.

2. Let the points 5,6,7 be the second intersections [= other than the point 4] of the lines 14, 24, 34 with the circles (423),(431),(412) resp.

3. Let the points 5,6,7 be the second intersections [= other than the point 4] of the circles (1,14),(2,24),(3,34) with the circles (423),(431), (412) resp.

4. Let 1',2',3' be the centers of the circles (423),(431),(412), resp. and the points 5,6,7, the second intersections [= other than the point 4] of the circles (42'3'),(43'1'),(41'2') with the circles (423),(431),(412) resp.

Continued 5

## 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