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Re: [EMHL] Ismail Erciyes' question

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• My formulas are rather lengthy (and i d not liked to give them, but as comparison and check with Barry Wolk s formula they may be of some sense); let: x[B,C]:=
Message 1 of 7 , Aug 7, 2003
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My formulas are rather lengthy
(and i'd not liked to give them,
but as comparison and check
with Barry Wolk's formula
they may be of some sense);

let:
x[B,C]:=
R ((1/2)Sqrt[Sin[2B]*Sin[2C]]-
Cos[B]*Cos[C]);
R is circumradius;
then:
areaDEF=
x[A,B]*x[A,C]*Sin[A]+
x[B,A]*x[B,C]*Sin[B]+
x[C,B]*x[C,A]*Sin[C];
%%%%%%%%%%
two numerical cases:
{a, b, c} = {4, 5, 6} :
areaDEF/areaABC =
3 (-45 + 30 Sqrt[7] - 22 Sqrt[15] +
7 Sqrt[105])/320 Sqrt[7] = 0.0740417
%%%%%%%%
{a, b, c} = {13, 14, 15} :
areaDEF/areaABC =
(3/236600)* (154 (120 - 7 Sqrt[5]) -
23 Sqrt[154](5 + 7 Sqrt[5])) = 0.129013.

Barry's formula gives (happily)
the same numerical values
Zak

--- In Hyacinthos@yahoogroups.com,
Barry Wolk <wolkbarry@y...> wrote:
> --- Darij Grinberg <darij_grinberg@w...> wrote:
> > Given an acute-angled triangle ABC, the A-altitude intersects
> > the circle with diameter BC at the point D. (In fact, there
> > are two intersections, but we take the one which lies on the
> > same half-plane of the line BC as A.) Analogously define the
> > points E and F.
> >
> > What is area(ABC) : area(DEF) ?
>
> Let x = (1-cot B)(1-cot C) / sqrt((cot B)(cot C)),
> with y and z defined cyclically. Then
> area(DEF) /area(ABC) = (cos A)(cos B)(cos C)(2-x-y-z)
>
> Routine barycentrics gave an answer, but reducing that answer to
> a short formula was tricky.
>
> --
> Barry Wolk
>
>
> __________________________________
> Do you Yahoo!?
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• Bernard, This ellipse is mentioned in Kapetis, but lightly, showing a nice method by which the axes of an inconic may be constructed. I have always liked this
Message 2 of 7 , Aug 10, 2003
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Bernard,

This ellipse is mentioned in Kapetis, but lightly, showing a nice
method by which the axes of an inconic may be constructed. I have
always liked this conic and did not know that the axes are parallel to
Jerabek. How did you decide this by the way?

Back after a long absence, friendly from USA,

Steve

>
> Dear friends,
>
> the in-conic with perspector X69 (isotomic of H) is an ellipse centered
> at O whose axes are parallel to the asymptotes of the Jerabek
> hyperbola. It contains X125 (center of the Jerabek hyperbola) and
> obviously its reflection E in O which is not mentionned in ETC.
>
> its 1st bary : S_A [ (b^2 - c^2)^2 + a^2 (b^2 + c^2 - 2a^2) ]^2
>
> the normals dropped from X20 to this ellipse pass through the cevians
> of X69 and E.
>
> two isogonal conjugate points have their polar lines in this ellipse
> which are parallel if and only if they lie on the McCay cubic.
>
> does someone have any reference about this ellipse ?
>
> Best regards
>
> Bernard
• Dear Steve, ... I expect it s a known result : for any point M on the Lucas cubic, the axes of the inconic with perspector M are parallel to the asymptotes of
Message 3 of 7 , Aug 11, 2003
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Dear Steve,

> This ellipse is mentioned in Kapetis, but lightly, showing a nice
> method by which the axes of an inconic may be constructed. I have
> always liked this conic and did not know that the axes are parallel to
> Jerabek. How did you decide this by the way?

I expect it's a known result :

for any point M on the Lucas cubic, the axes of the inconic with
perspector M are parallel to the asymptotes of the rectangular
circum-hyperbola through M.

Best regards

Bernard

PS : Antreas kindly sent me the Kapetis book. Unfortunately, I can't
read Greek.
Where do you find the information about this ellipse ?
Antreas, please provide a translation. Thanks.

[Non-text portions of this message have been removed]
• ... Dear Bernard Right now I don t see it in my chaotic library! Only the 2nd volume I bought recently. Antreas PS: When more copies arrive from Thessaloniki
Message 4 of 7 , Aug 12, 2003
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Bernard Post-Scripted:

>PS : Antreas kindly sent me the Kapetis book. Unfortunately, I can't
>read Greek.
>Where do you find the information about this ellipse ?
>Antreas, please provide a translation. Thanks.

Dear Bernard

Right now I don't see it in my chaotic library!
Only the 2nd volume I bought recently.

Antreas

PS: When more copies arrive from Thessaloniki
(the town where the book was published) in the bookstore,
then I will send you that volume too
APH

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