ADGEOM 833 - ADGEOM 852

 

#833

Dear friends:

Let P be a point in the plane of ABC, A’B’C’ the cevian triangle of ABC and Ia, Ib, Ic its excenters.

The lines (IaC’) cut (IaB’) cut (BC) at Ab and Ac, respectively. Build Ba, Bc, Ca, Cb cyclically.

Then:

1) The points Ab, Ac, Ba, Bc, Ca, Cb lie on a conic.

2) If P=u:v:w (trilinears) then the center of the conic is:

    Z(P) = 2*(b+c)*u+(c+2*b)*v+(2*c+b)*w-a*v*w/u : :

3) If P lies on circumcircle of ABC then Z(P)=X(37)

4) Z(P) lies on line (X(37)P*), where P*=isogonal conjugate of P

 

ETC pairs [P,Z(P)] = [X(I),X(J)] for these (I,J)=

(2,1001), (44,88), (57,1), (81,37), (88,1), (513,100), (649,190), (650,651),       

(652,653), (654,655), (656,162), (657,658), (659,660), (661,662), (672,673),       

(770,771), (798,799), (822,823), (896,897), (1155,1156), (1491,1492), (1635,3257), 

(1755,1821), (2173,2349), (2578,2580), (2579,2581), (3218,1807), (3768,4607),      

(4790,4606), (4893,4604)

 

Non-ETC:

Z( X(1) ) = 5*b + 5*c-a : :

    On lines (1,6), (2,1266), (8,4029), (10,4873), (63,89), (142,4346), (145,3707), (346,5257), 

 (527,5308), (966,3626), (968,3158), (1696,5217), (2321,3617),         

 (2345,3634), (3161,5550), (3241,4700), (3621,3686), (3625,4034), (3677,3989),      

 (3679,3943), (3729,4687), (3875,4704), (3929,5287), (4363,4755), (4384,4664),      

 (4419,4887), (4648,4896), (4677,4727)

 

Z( X(6) ) = (b+c)*a+b^2+b^2+(b*c)/2  : :

   On lines (1,89), (2,37), (8,4868), (38,4430), (42,4661), (43,3989), (63,1449), (145,3931),  

 (982,1962), (984,3240), (1150,4360), (1743,3219), (1999,5372),         

 (3187,5361), (3247,3306), (3616,3743), (3936,4389), (4361,5235), (4646,4678),      

 (4651,4734), (4709,4970)

 

Regards

César Lozada

 

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#842

Dear  César,

This may be the same conic discussed in Hyacinthos #20547.  Your point Z(X(1)) matches a point I have in my notes as 'Hyacinthos #20547 [https://aphyacinthos.blogspot.com/2019/10/hyacinthos-20547.html] conic center when P = X(1)', but the new Yahoo groups format makes it difficult to find a message based on message #, so I cannot confirm.

 

Best regards,

Randy Hutson

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#844

Thank you, Francisco Javier.  These two conics are similar: yours uses the cevian and anticevian triangles of a point P, while César's uses the cevian triangle of P and the anticevian triangle of X(1).

Reviewing my notes from back then, I had found a (unique?) point P such that the conic (
Francisco Javier's) is a circle.  It is a non-ETC center, P^ (search=0.834038702291580), as is the center, O^ (search=2.227553680841584).  P^ and O^ are collinear with X(3534), but I could find nothing else of interest about these points, except that P^ is also the perspector of the circle.  (Does this hold in general?)

 

I wonder which is the (unique?) point P such that César's conic is a circle.

 

Best regards,

Randy Hutson

 

--Francisco Javier wrote:

Dear Randy, according to my record, this is Hyacinthos #20547:

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From: "Francisco Javier" <garciacapitan@...>
Date: Mon Dec 19, 2011 10:47 am
Subject: The anticevian intersections conic     garciacapitan
  
Dear friends,

As a little Christmas present, in

http://garciacapitan.blogspot.com/2011/12/anticevian-intersection-conic.html

you can read a little research on a conic through six intersection points.

I gladly wish to express my gratitude to Jean-Pierre Ehrmann, Bernard Gibert and Sung Hyun Lim for several helpful comments concerning this problem.

Have a Happy Christmas and New Year 2012.

Francisco Javier García Capitán.

--------------------------------------------

 

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#848

Another interesting property of P^: it is the trilinear pole of its polar wrt circle (O^).  Similar to K wrt circumcircle.

Coordinates?

Best regards,
Randy Hutson

 

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#852

[Randy Hutson]:

>> I wonder which is the (unique?) point P such that César's conic is a circle.

[César Lozada]:

The conic is a circle for P with ETC-search 1.41380272606812.., 3.85478662709369.., 0.319441712657307.. (at least)

The center Z of the circle for this P has ETC-search 5.302845236905069.., 5.39706500845055.., -2.543232171745264..

No relations were found between P, Z and other ETC centers.

Best regards

César Lozada