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X(5415) & X(5416) coordinates

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• Dear geometers: Here are two coordinates pending in ETC. Maybe this information may be of interest for Professor Kimberling. Regards César Lozada. X(5415) =
Message 1 of 3 , Apr 6, 2013
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Dear geometers:

Here are two coordinates pending in ETC.

Maybe this information may be of interest for Professor Kimberling.

Regards

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]
• ... what is the meaning of major center ? aph [Non-text portions of this message have been removed]
Message 2 of 3 , Apr 7, 2013
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On Sun, Apr 7, 2013 at 7:00 AM, César Lozada <cesar_e_lozada@...>wrote:

> **
>
>

>
> X(5415) is a major center.
>
>
> X(5416) is a major center.
>

what is the meaning of "major center"?

aph

[Non-text portions of this message have been removed]
• Very nice, César! ... 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
Message 3 of 3 , Apr 7, 2013
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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

wrote:
>
>
> Dear geometers:
>
>
> Here are two coordinates pending in ETC.
>
>
> Maybe this information may be of interest for Professor Kimberling.
>
>
> Regards
>
>
>
>
>
>
>
> 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]
>
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