- Dear Hyacinthists,

Are these known results?

1) Let Ea be the ellipse with B and C as foci and passing through A. Define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.

2) Let A', B', C' be points on segments BC, CA, AB respectively. Let Ea be the ellipse with B' and C' as foci and passing through A, and define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.

3) Let P and P' be isogonal conjugates both inside ABC. Let Ea be the ellipse with P and P' as foci and passing through A, and define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.

Thank you,

Randy Hutson - Closely related to this: If 'hyperbola' is substituted for 'ellipse' below, then in each case, the tangents concur in the incenter.

Randy

>________________________________

[Non-text portions of this message have been removed]

> From: rhutson2 <rhutson2@...>

>To: Hyacinthos@yahoogroups.com

>Sent: Sunday, July 29, 2012 10:51 PM

>Subject: [EMHL] excentral triangle as tangents to ellipses

>

>

>

>Dear Hyacinthists,

>

>Are these known results?

>

>1) Let Ea be the ellipse with B and C as foci and passing through A. Define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.

>

>2) Let A', B', C' be points on segments BC, CA, AB respectively. Let Ea be the ellipse with B' and C' as foci and passing through A, and define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.

>

>3) Let P and P' be isogonal conjugates both inside ABC. Let Ea be the ellipse with P and P' as foci and passing through A, and define Eb, Ec cyclically. Let La be the tangent to Ea at A, and define Lb, Lc cyclically. The lines La, Lb, Lc form the excentral triangle.

>

>Thank you,

>Randy Hutson

>

>

>

>

>

- Dear Randy

I think that are known

because if F, F' are the foci

of an ellipse and P a point on it

then the tangent at P to the ellipse

is perpendicular to the internal bisector PQ

of triangle PFF' at P.

In case of an hyperbola the tangent is the line PQ.

Best regards

Nikos Dergiades

> Are these known results?

>

> 1) Let Ea be the ellipse with B and C as foci and passing

> through A. Define Eb, Ec cyclically. Let La be

> the tangent to Ea at A, and define Lb, Lc cyclically.

> The lines La, Lb, Lc form the excentral triangle.

>

> 2) Let A', B', C' be points on segments BC, CA, AB

> respectively. Let Ea be the ellipse with B' and C' as

> foci and passing through A, and define Eb, Ec

> cyclically. Let La be the tangent to Ea at A, and

> define Lb, Lc cyclically. The lines La, Lb, Lc form

> the excentral triangle.

>

> 3) Let P and P' be isogonal conjugates both inside

> ABC. Let Ea be the ellipse with P and P' as foci and

> passing through A, and define Eb, Ec cyclically. Let

> La be the tangent to Ea at A, and define Lb, Lc

> cyclically. The lines La, Lb, Lc form the excentral

> triangle.

>

> Thank you,

> Randy Hutson

>

>

>

> ------------------------------------

>

> Yahoo! Groups Links

>

>

> Hyacinthos-fullfeatured@yahoogroups.com

>

> - Dear Nikos,

Of course, this makes perfect sense now. Thanks!

Randy

>________________________________

[Non-text portions of this message have been removed]

> From: Nikolaos Dergiades <ndergiades@...>

>To: Hyacinthos@yahoogroups.com

>Sent: Tuesday, July 31, 2012 12:42 AM

>Subject: Re: [EMHL] excentral triangle as tangents to ellipses

>

>Dear Randy

>I think that are known

>because if F, F' are the foci

>of an ellipse and P a point on it

>then the tangent at P to the ellipse

>is perpendicular to the internal bisector PQ

>of triangle PFF' at P.

>In case of an hyperbola the tangent is the line PQ.

>Best regards

>Nikos Dergiades

>

>> Are these known results?

>>

>> 1) Let Ea be the ellipse with B and C as foci and passing

>> through A. Define Eb, Ec cyclically. Let La be

>> the tangent to Ea at A, and define Lb, Lc cyclically.

>> The lines La, Lb, Lc form the excentral triangle.

>>

>> 2) Let A', B', C' be points on segments BC, CA, AB

>> respectively. Let Ea be the ellipse with B' and C' as

>> foci and passing through A, and define Eb, Ec

>> cyclically. Let La be the tangent to Ea at A, and

>> define Lb, Lc cyclically. The lines La, Lb, Lc form

>> the excentral triangle.

>>

>> 3) Let P and P' be isogonal conjugates both inside

>> ABC. Let Ea be the ellipse with P and P' as foci and

>> passing through A, and define Eb, Ec cyclically. Let

>> La be the tangent to Ea at A, and define Lb, Lc

>> cyclically. The lines La, Lb, Lc form the excentral

>> triangle.

>>

>> Thank you,

>> Randy Hutson

>>

>>

>>

>> ------------------------------------

>>

>> Yahoo! Groups Links

>>

>>

>> Hyacinthos-fullfeatured@yahoogroups.com

>>

>>

>

>

>------------------------------------

>

>Yahoo! Groups Links

>

>

>

>

>

>