Let ABC be a triangle and AA', BB', CC' its altitudes. Let Ab, Ac, Bc, Ba, Ca, Cb be the projections of A', A', B', B', C', C' on AB, CA, BC, AB, CA, BC,...
Dear Darij in Hyacinthos 7408, you wrote ... Note that if an angle of ABC is obtuse, the incenter of A'B'C' is not H, but the corresponding vertex of ABC. ... ...
Dear Jean-Pierre Ehrmann, ... Yes, of course I know this, the excenters of a triangle also lie on the Neuberg cubic. ... I have met this before. The point M...
Dear Darij [DG] ... [JPE] ... is ... [DG] ... I was just meaning that the common point is the Schiffler point of A'B'C' only when ABC is acutangle ... I'm...
We begin with some definitions related to a given triangle ABC. A triangle A'B'C' is called JACOBI TRIANGLE if angle C'AB = angle CAB'; angle A'BC = angle...
Dear Darij, here are some elements of answer. To any point P, we can associate an unique special Schaal triangle such as the lines AA',BB',CC' and the circles...
Dear Jean-Pierre Ehrmann, ... Thanks, of course, I was stupid enough not no note this myself. (I was half asleep when I wrote the mail.) So let me, at least,...
Dear Darij ... If, for a special Schaal triangle, we call P the central point of the triangle then - A'*B'*C'* is a special Schaal triangle with central point...
Re: Generalizing "On the Kosnita Point and the Reflection Triangle" Dear Jean-Pierre Ehrmann, ... Indeed, if A'*, B'*, C'* are the isogonal conjugates of A',...
Dear all, ... I think that a synthetic proof (in Russian) can be found at http://archive.1september.ru/mat/2000/no43_1.htm Alas, I cannot follow the author's...
Dear Jean-Pierre Ehrmann, ... Let me start with the proof of this assertion and of A'A'* || B'B'* || C'C'* || PP*. I will use directed angles modulo 180°. The...
In my note "On the Kosnita Point and the Reflection Triangle", Forum Geometricorum 3 (2003) pages 105-111, I studied the "reflection triangle" A'B'C' of a ...
... Here is a PROOF. Let Rc and Ra be the circumcenters of triangles ABC' and BCA', respectively, i. e. the reflections of the circumcenter O of triangle ABC...
Take any triangle ABC, and point P inside it. Three reflections of P in sides AB, BC, and AC give three vertices of "reflection triangle" DEF. For which P, the...
Dear Hyacinthists does anybody know the first appearance of the following result - may be in a Hyacinthos message? - : A'B'C' = cevian triangle of P Q is the...
Here is a little foundling:- Emil Donath, "Die merkwürdigen Punkte und Linien des ebenen Dreiecks" on page 52: "Es ist ohne weiteres klar, daß Dreieck ASB2 ...
Dear Zak Seidov, ... There are two such points: the "isodynamic points" X(15) and X(16). If triangle ABC is acute, then one of them is inside triangle ABC and...
Dear Hyacinthos, I am looking for the books about open problems in geometry, especially those, which formulating can be explained to students with no special...
Ismail Erciyes has just tried to send a problem to Hyacinthos but didn't succeed. Dear Ismail, unfortunately attachments are not possible in YahooGroups. Let...
Dear Darij as, for every point M, <BMC + <BM*C = <BAC, the circle BCP* is the isogonal conjugate of the circle BCP. It follows immediately that, if A'B'C' is a...
Dear Darij, in one of your's messages you wrote ... .SYNTHETIC proof? The proof can be found in Sharygin's book of problems from elementary geometry. The...
Dear Jean-Pierre, ... Yes, I knew this. Together with the collinearity of A, P* and A'*, this yields that A'*B'*C'* is a special Schaal triangle with central...
Dear Darij, let's try a generalization of the property of the circumcenters. I suppose that P lies on the Neuberg cubic; in this case the triangle with...
Dear Darij another property of the nice configuration you've discovered (for any point P) The polar conic of the infinite point F* of PP* wrt the self isogonal...
... 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...
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....
Let me start by saying I'm far from a mathematician, however this question was posed to me many years ago in high school and I was never able to find a...
