- 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 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]

> >[JS]

Dear John,

> 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

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