- Dear geometers:

Here are two coordinates pending in ETC.

Maybe this information may be of interest for Professor Kimberling.

Regards

César Lozada.

X(5415) = 4th KENMOTU HOMOTHETIC CENTER

Trilinears (pending)

The A'B'C' defined at X(5412) is homothetic to the extangents triangle, and

the center of homothety is X(5415). Also, X(5415) = {X(6), X(55)}-harmonic

conjugate of X(5416). (Randy Hutson, February 9, 2013)

The coordinates of X(5415) are:

Trilinears: a*( (2*R + r ) + (s-a) ) : :

Trilinears: f(A) : f(B) : f(C) , where f(A)=( (2*R + r + s)/R - 2*sin(A)) *

sin(A)

Barycentrics f(A)*sin(A) : f(B)*sin(B) : f(C)*sin(C)

ETC SEARCH : 1.246310683967065.., 1.64076433091358.., 1.929530398674793..

X(5415) is a major center.

X(5415) lies on lines : X(6)-X(31), X(19)-X(5412) , X(40)-X(371)

X(5416) = 5th KENMOTU HOMOTHETIC CENTER

Trilinears (pending)

The A'B'C' defined at X(5413) is homothetic to the extangents triangle, and

the center of homothety is X(5415). Also, X(5416) = {X(6), X(55)}-harmonic

conjugate of X(5416). (Randy Hutson, February 9, 2013)

The coordinates of X(5416) are:

Trilinears: a*( (2*R + r ) (s-a) ): :

Trilinears: f(A) : f(B) : f(C) , where f(A) = ( (2*R + r - s)/R +

2*sin(A))*sin(A)

Barycentrics f(A)*sin(A) : f(B)*sin(B) : f(C)*sin(C)

ETC SEARCH : 0.626431598880468.., 1.27040179200258.., 2.472033272576223..

X(5416) is a major center.

X(5416) lies on lines : X(6)-X(31), X(19)-X(5413) , X(40)-X(372)

[Non-text portions of this message have been removed] - On Sun, Apr 7, 2013 at 7:00 AM, César Lozada <cesar_e_lozada@...>wrote:

> **

what is the meaning of "major center"?

>

>

>

> X(5415) is a major center.

>

>

> X(5416) is a major center.

>

aph

[Non-text portions of this message have been removed] - Very nice, César!

I had figured the trilinears might be in the form 1 + k sin A + cos A :

: for X(5415) and 1 - k sin A + cos A : : for X(5416), for some

appropriate value of k (for the ETC reference triangle, it appears to be

391/40), but I could never find a function for k. I wonder if there are

related trilinears in your form for X(2066) and X(5414).

Antreas, to answer your question (and I had to look it up myself to

remember), a major center is one whose triangle center function alpha is

a function of angle A (or side a) alone. Since César's function

contains R, r and s, which themselves are functions of a, b, and c, I

wonder if they technically qualify as major centers.

Best regards,

Randy

--- In Hyacinthos@yahoogroups.com, César Lozada <cesar_e_lozada@...>

wrote:>

triangle, and

>

> Dear geometers:

>

>

> Here are two coordinates pending in ETC.

>

>

> Maybe this information may be of interest for Professor Kimberling.

>

>

> Regards

>

>

> César Lozada.

>

>

>

>

>

> X(5415) = 4th KENMOTU HOMOTHETIC CENTER

>

>

> Trilinears (pending)

>

> The A'B'C' defined at X(5412) is homothetic to the extangents

> the center of homothety is X(5415). Also, X(5415) = {X(6),

X(55)}-harmonic

> conjugate of X(5416). (Randy Hutson, February 9, 2013)

2*sin(A)) *

>

> The coordinates of X(5415) are:

>

> Trilinears: a*( (2*R + r ) + (s-a) ) : :

>

> Trilinears: f(A) : f(B) : f(C) , where f(A)=( (2*R + r + s)/R -

> sin(A)

1.929530398674793..

>

> Barycentrics f(A)*sin(A) : f(B)*sin(B) : f(C)*sin(C)

>

>

>

> ETC SEARCH : 1.246310683967065.., 1.64076433091358..,

>

triangle, and

>

>

> X(5415) is a major center.

>

> X(5415) lies on lines : X(6)-X(31), X(19)-X(5412) , X(40)-X(371)

>

>

>

>

>

>

> X(5416) = 5th KENMOTU HOMOTHETIC CENTER

>

>

> Trilinears (pending)

>

> The A'B'C' defined at X(5413) is homothetic to the extangents

> the center of homothety is X(5415). Also, X(5416) = {X(6),

X(55)}-harmonic

> conjugate of X(5416). (Randy Hutson, February 9, 2013)

2.472033272576223..

>

> The coordinates of X(5416) are:

>

> Trilinears: a*( (2*R + r ) (s-a) ): :

>

> Trilinears: f(A) : f(B) : f(C) , where f(A) = ( (2*R + r - s)/R +

> 2*sin(A))*sin(A)

>

> Barycentrics f(A)*sin(A) : f(B)*sin(B) : f(C)*sin(C)

>

>

>

> ETC SEARCH : 0.626431598880468.., 1.27040179200258..,

>

> X(5416) is a major center.

>

> X(5416) lies on lines : X(6)-X(31), X(19)-X(5413) , X(40)-X(372)

>

>

>

>

>

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

>