Dear Bernard! Thank you Alexey Dear Alexey, ... see X(1144) = EHRMANN CONGRUENT SQUARES POINT best regards Bernard [Non-text portions of this message have been...
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Nikolaos Dergiades
ndergiades
Nov 6, 2012 6:02 pm
Dear Alexey, this point X(1144) as Bernard mentioned as solution to your problem, lies on the conic A,B,C,X(1),X(6) and on the hyperbola that passes through A...
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Nikolaos Dergiades
ndergiades
Nov 6, 2012 10:29 pm
Dear friends, Let A'B'C' be the cevian triangle of a point P. Let L1 be the conic with axis the perpendicular bisector of AP and that passes through A,P,B',C'....
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Bernard Gibert
bernardgibert
Nov 7, 2012 8:07 am
Dear Nikolaos, ... I should say yes since it is the Neuberg cubic... best regards Bernard [Non-text portions of this message have been removed]...
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Antreas Hatzipolakis
xpolakis
Nov 7, 2012 9:31 am
... ..... and the other components?? aph ... [Non-text portions of this message have been removed]...
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Bernard Gibert
bernardgibert
Nov 7, 2012 9:39 am
Dear Antreas, ... the circles with diameters BC, CA, AB the line at infinity the sidelines of ABC not very interesting... best regards Bernard ... [Non-text...
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Nikolaos Dergiades
ndergiades
Nov 7, 2012 4:52 pm
Dear Bernard [BG] ... Yes. This is the locus. As for the cubic although I found the correct result I made I mistake finding the pivot of the cubic and thought...
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César Lozada
cesar_e_lozada
Nov 8, 2012 7:29 pm
Dear geometers: Let P be a point not the orthocenter and not on the circumcircle. Let Pa, Pb, Pc be the reflections of P on the sides of triangles. Then the...
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CESAR
cesar_e_lozada
Nov 8, 2012 7:32 pm
Sorry: I meant: Excluding cases 1-3 from above.... Thank you....
Dear Cesar, ... As Randy Hutson said Q is the antigonal conjugate of P and to find T of Euler line see at K025 (Ehrmann Strophoid) of Bernard Gibert katalogue ...
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CESAR
cesar_e_lozada
Nov 9, 2012 3:40 pm
Dear Randy & Nikos: You are absolutely right. Thank you. César Lozada...
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rhutson2
Nov 9, 2012 11:14 pm
Dear friends, Consider the mapping F(P), defined equivalently as: 1) F(P) = pole, wrt polar circle, of the trilinear polar of P 2) F(P) = trilinear pole of the...
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Angel
amontes1949
Nov 12, 2012 2:51 pm
Dear Hyacinthists. (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143) Given triangle ABC, consider the...
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Randy Hutson
rhutson2
Nov 13, 2012 3:07 am
Dear Angel, How does the A-Soddy hyperbola have a perspector if it is not a circumconic? Randy ... [Non-text portions of this message have been removed]...
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Angel
amontes1949
Nov 13, 2012 9:06 am
... Dear Randy Hutson, Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 10.7.3 The perspector of a conic The Soddy hyperbolas, p. 122 Theorem...
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Francisco Javier Garc...
garciacapitan
Nov 13, 2012 11:00 am
Note that this is not true for a general conic: the perspector of a conic, i.e. the perspector of triangle ABC and the triangle formed by the polars of A, B, C...
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Angel
amontes1949
Nov 13, 2012 1:27 pm
Dear Hyacinthists, Let ABC be a triangle, D a point on AC, E a point on AB such that DE=BC=a. The locus of the centroids of the triangle ADE is a conic (Ca)...
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Randy Hutson
rhutson2
Nov 13, 2012 1:49 pm
Dear Angel and Francisco, Thank you very much. That makes sense now. I guess I need to make the time to finish reading Yiu's book. One thing that I think...
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Randy Hutson
rhutson2
Nov 13, 2012 5:30 pm
Never mind. I see now that even though one vertex of the 'polar triangle' is on the line at infinity, the triangle is still perspective to ABC. Randy ... ...
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rhutson2
Nov 13, 2012 8:08 pm
Dear Angel, If you substitute ellipses for hyperbolas in this construction, the resulting polars are not concurrent, but form a triangle which is perspective...
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John Sharp
sliceforms
Nov 13, 2012 8:56 pm
Mention of the Soddy hyperbolae made me construct them in Geogebra Does anyone know of the following results ie a reference in the literature: 1) The three...
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rhutson2
Nov 13, 2012 9:14 pm
Dear John, I count 6 intersections (at least with the ETC reference triangle). It may depend on the shape of the triangle. I wonder if these 6 points lie on...
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Chris Van Tienhoven
chris.vantie...
Nov 13, 2012 10:07 pm
... Dear John, I know these refences. 1. Both points are briefly mentioned at: http://www.xtec.cat/~qcastell/ttw/ttweng/definicions/d_Soddy_p.html 2. There is...
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John Sharp
sliceforms
Nov 14, 2012 2:25 pm
Dear Randy & Chris Thanks for the response on my observations on the Soddy Ellipses & hyperbolae Randy, I did miss the 6th intersection, even though I drew the...
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rhutson2
Nov 14, 2012 6:44 pm
Dear friends, I received word recently of the passing of Richard L. "Rick" Parris in late October. Rick was a professor of Mathematics at Phillips Exeter...
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rhutson2
Nov 16, 2012 12:12 am
Question: Is there a construction of this mapping that does not involve the polar circle, which is only defined for obtuse triangles? Randy...
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Bernard Gibert
bernardgibert
Nov 16, 2012 10:16 am
Dear Randy, ... your F(P) is the H-isoconjugate of P i.e. the image of P under the isoconjugation that swaps G and H. best regards Bernard [Non-text portions...
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Bernard Gibert
bernardgibert
Nov 20, 2012 3:36 pm
... Dear Sung Hyun, I've done quite a lot of research on this subject recently and I was wondering whether somebody else has already done the same since...
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rhutson2
Nov 20, 2012 9:44 pm
Dear Bernard, After looking at it more closely, it looks like F(P) is the X(48)-isoconjugate of P, or equivalently, the trilinear product of X(92) and the...
