An interesting thread was started in the geometry-college group: http://mathforum.org/epigone/geometry-college/pringkheldeh ... Subject: Brocard points of...
Let ABC be a triangle, P a point and PaPbPc the pedal triangle of P. Denote Pab = The Orth. Projection of Pa on CA Pac = The Orth. Prpoection of Pa on AB Pbc =...
Dear Daarij and Eric, [DG]: ... Now, consider a triangle ABC, and the isogonal conjugate R of its t-Kiepert perspector. Then, my conjecture says that the point...
Dear Antreas ... If the normal coordinates of P are x:y:z, then the signed distance PaPab is y + x cos(C) and the signed distance PaPac is z + x cos(B). Thus...
Dear Bernard and Antreas, [BG]: consider the circumcenters O1 and O2 of the pedal and antipedal triangles of a point P. For which P do O1 and O2 coincide ? ***...
Dear Antreas ... If the normal coordinates of P are x:y:z, then the signed distance PaPab is y + x cos(C) and the signed distance PaPac is z + x cos(B). Thus...
Darij raised some interesting questions about harmonic quadrilaterals. ... Casey, 'A sequel to Euclid', has various theorems on harmonic n-gons, including a...
dick tahta
dick@...
Jun 1, 2004 4:13 pm
9831
The Brocard points of a (harmonic) quadrilateral are mentioned by John Casey in 'A Sequel to Euclid'. I have the comparatively rare fifth edition (published by...
... Dear Jean-Pierre Another locus is this: Locus of P such that the triangles bounded by the lines (PaPab , PbPbc , PcPca) and (PaPac , PcPcb , PbPba) are...
Dear Dick, ... Thank you very much for the information. My dynamic sketch confirms csc^2 w = csc^2 A + csc^2 B for a harmonic quadrilateral. But what is the...
Dear Paul, In Hyacinthos message #9826, you wrote: [...] ... This is very nice! At least, it is clear that triangle XB'C' isosceles (for the fundamental ...
... Was Third's reference earlier than 1949? I remember striking this problem as part of a longer proof of an old text-book exercise: E. A. Maxwell, "Geometry...
Ken Pledger
Ken.Pledger@...
Jun 1, 2004 9:55 pm
9837
Dear Antreas ... These triangles are allways congruent - infact, reflected of each other wrt a point - ... The locus of P such as the 6 points lie on a conic...
Dear Ken Pledger, Thanks for the information in Hyacinthos message #9836. Lately, I have succeeded to localise the Third reference: J. A. Third, "Systems of...
Dear Geoff Millin and Darij! ... This results are very interesting. I have one question. It is known that the projections of two isogonally conjugated points...
Alexey.A.Zaslavsky
zasl@...
Jun 2, 2004 10:10 am
9843
Dear Alexey, ... This is very interesting! At first, consider the Brocard points R and R' of the harmonic quadrilateral ABCD, and the symmedian point K (this...
Dear Antreas, if PaPbPc is the pedal triangle of P, Ma = common point of the perpendicular at Pa to AC and the perpendicular at Pb to AB, Mb, Mc cyclic Na =...
In the past of the Hyacinthos newsgroup, several proofs were presented for the so-called Droz-Farny theorem. Here is one possible formulation of this theorem: ...
Darij, Paul, a.o. This is the circucmcevian summary of Darij, 16/5/2004. ______________________________ Let P be a point in the plane of a triangle ABC, and...
Dear Clark Below JPE gives a new point with barycentrics (cos^2B + cos2^C ::) You might want to add it in ETC with the name EHRMANN CIRCLE CENTER Greetings ...
Dear Jean-Pierre ... I think that it is better to say "the perpendicular FROM [point] TO [line]" But my English is not perfect either! Thanks for the nice...
Dear Aad, ... Thanks a lot for this observation! It really clarifies a lot. Now I can simplify your proof of (3). I won't use any trigonometry, but I will use ...
Dear Hyacinthists, we are now very close from 10000 messages in Hyacinthos. This is a nice occasion for a little problem of construction; I hope that you will...
It regularly happens in mathematics that someone poses a conjecture, it remains open for a long time, until someone else finds a long and elaborate proof, then...
Treegoner posted a neat triangle problem in the MathLinks thread http://mathlinks.ro/viewtopic.php?p=20005 Here the problem goes: Let (O) be a fixed circle and...
Dear Jean-Pierre ... I haven't tried a construction yet but I have proven that your four lines are the trilinear polars of four (two by two isogonal) points on...
Dear Darij! ... I can prove some of this facts. Let P is the first Brocard point and the lines AP, BP, CP, DP intersect the circumcircle in B', C', D', A'....
Alexey.A.Zaslavsky
zasl@...
Jun 4, 2004 7:19 am
9855
Dear Darij! ... There is one correction. P is the midpoint of KK' because OK=OK'. This follows from next theorem. Let four chords A_iB_i concur in point C. D...
Alexey.A.Zaslavsky
zasl@...
Jun 4, 2004 10:34 am
9856
Dear colleagues! Does anybody know the non projective proof of next fact. Let A'B'C' is the circumcevian triangle of ABC with respect to point P, L, L' are the...
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