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Clifford's circle chain theorem with n=8 circles

For clarity you first see the **central part** of the drawing and next a **global view**
including all points and circles.

Following Penrose' notation, I omit letters "c" for circles and "P" for points,
to avoid cluttering.
Here we see eight (n=8)
blue circles c1, c2, c3, c4, c5, c6, c7, c8 through a point O (taken to infinity)
intersecting in
P12, P13, P14, P15, P16, P17, P23, ..., P78
which again define
unique yellow circles like as c123, c124, ..., c678 (not all drawn, to
avoid cluttering).
The construction of these yellow circles corresponds to applying the case
n=3
successively to three circles each, i.e. to c1,c2,c3 to give c123, etc.
According to the case with
n=4
circles through O taking four yellow circles at a time yields
the gold colored points P1234, P1235, ..., P4567.
The case n=5 shows that always five of the golden points
at a time, e.g. P1234, P1235, P1245, P1345, P2345 lie on red
circles c12345, c12346, ..., c45678.
According to the case with
n=6
allways six such red circles, e.g. c12345,
c12346, c12356, c12456, c13456, c23456 will coincide in (violet) points like P123456,
P123457, ..., P345678.
The case with
n=7
showed that always seven such points, e.g.
P123456, P134567, P124567, P123567, P123467, P123457, P123456 lie on one
unique (violet) circles
1234567, and c1234568, c1234578, c1234678, c1235678, c1245678, c1345678, c2345678.
The surprising fact shown below with n=8 initial blue circles is, that the final
eight violet circles intersect in one unique (green) point P12345678, which is
best seen in the global view below.

The bright red points can be moved interactively with the mouse pointer.
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Clifford's circle chain theorem with n=8 circles - global view

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[ circle chain theorem | GA with Cinderella ]

Soli Deo Gloria. Created with Cinderella
by Eckhard Hitzer (Fukui).