- 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!?

> Yahoo! SiteBuilder - Free, easy-to-use web site design software

> http://sitebuilder.yahoo.com - 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,

> This ellipse is mentioned in Kapetis, but lightly, showing a nice

I expect it's a known result :

> 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?

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] - Bernard Post-Scripted:

>PS : Antreas kindly sent me the Kapetis book. Unfortunately, I can't

Dear Bernard

>read Greek.

>Where do you find the information about this ellipse ?

>Antreas, please provide a translation. Thanks.

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

--