## Soddy hyperbolae & ellipses

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• Mention of the Soddy hyperbolae made me construct them in Geogebra Does anyone know of the following results ie a reference in the literature: 1) The three
Message 1 of 3 , Nov 13, 2012
Mention of the Soddy hyperbolae made me construct them in Geogebra

Does anyone know of the following results ie a reference in the literature:

1) The three hyperbolae all intersect in two points. Using the ETC, these
are
X(175) = ISOPERIMETRIC POINT
X(176) = EQUAL DETOUR POINT

2) Constructing the corresponding ellipses (two vertices as foci + the
other as a point)

These three ellipses intersect in 5 points which are also on the hyperbolae
These points are not in the ETC

John S

[Non-text portions of this message have been removed]
• Dear John, I count 6 intersections (at least with the ETC reference triangle). It may depend on the shape of the triangle. I wonder if these 6 points lie on
Message 2 of 3 , Nov 13, 2012
Dear John,

I count 6 intersections (at least with the ETC reference triangle). It may depend on the shape of the triangle. I wonder if these 6 points lie on a common conic.

Awhile back, I had found that the lines connecting the intersections of pairs these ellipses concur in X(20).

Best regards,
Randy

--- In Hyacinthos@yahoogroups.com, John Sharp <JS.sliceforms@...> wrote:
>
> Mention of the Soddy hyperbolae made me construct them in Geogebra
>
> Does anyone know of the following results ie a reference in the literature:
>
> 1) The three hyperbolae all intersect in two points. Using the ETC, these
> are
> X(175) = ISOPERIMETRIC POINT
> X(176) = EQUAL DETOUR POINT
>
> 2) Constructing the corresponding ellipses (two vertices as foci + the
> other as a point)
>
> These three ellipses intersect in 5 points which are also on the hyperbolae
> These points are not in the ETC
>
> John S
>
>
> [Non-text portions of this message have been removed]
>
• ... Dear John, I know these refences. 1. Both points are briefly mentioned at: http://www.xtec.cat/~qcastell/ttw/ttweng/definicions/d_Soddy_p.html 2. There is
Message 3 of 3 , Nov 13, 2012
>[JS]
> Does anyone know of the following results ie a reference in the literature:
>
> 1) The three hyperbolae all intersect in two points. Using the ETC, these
> are
> X(175) = ISOPERIMETRIC POINT
> X(176) = EQUAL DETOUR POINT

Dear John,

I know these refences.
1. Both points are briefly mentioned at: http://www.xtec.cat/~qcastell/ttw/ttweng/definicions/d_Soddy_p.html
2. There is a more extensive description at this site:
http://www.pandd.demon.nl/
choose: M E E T K U N D E (at the left)
choose: S
choose: Soddy cirkels
The site is in the Dutch language.

Some years ago I found a most peculiar property of 10 Soddy-related points, including X(175) and X(176).
These 10 points X(1), X(176), X(1371), X(482), X(1373), X(7), X(1374), X(481), X(1372), X(175) are collinear on the X(1).X(7)-line.
They always lie in this order.
They lie in a "perspective row" with vanishing point X(1).
I made a picture of it in the file "Perspective Fields - part II" page 31.
See: http://www.chrisvantienhoven.nl/index.php/mathematics/perspective-fields.html
An explanantion of this feature can be found in the rest of the file.

Best regards,

Chris van Tienhoven
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