Dear Hyacinthists, ABC and A'B'C' are two given triangles. Do you know a way to construct the pair (M,N) of points such as M,N are isogonal conjugates wrt ABC...
... Here is a little idea: Call M the midpoint of IbIc, then OM = R. Let La be the pedal of I on B'C', and X be the pedal of I on BC, then IX = r. To prove...
On the Mathworld pages, I see a topic on a "Yff Hyperbola": "This hyperbola has vertices at the triangle centroid G and orthocenter H, a focus at the...
I heard that the orthocenter was one of the triangle centers known to the ancient greeks. Does anyone know who is attributed with the discovery and what greek...
Given a triangle ABC, let the perpendicular bisector of BC intersect AB at Ac, and the perpendicular bisector of AB intersect BC at Ca. The intersection Tb of...
In the 46th issue of "mathematik lehren" (a German math. teacher magazine), I found an article "Eine interessante geometrische Aufgabe" of Günter Grosche. He...
In the article Hans Walser, Ein Schnittpunktsatz, Praxis der Mathematik 2/1991 pages 70-71, the following theorem is proven (I call it Walser Theorem): Given a...
Dear Hyacinthians, Let ABC be a triangle, G its Gergonne point, N its Nagel point, M its Mitten point, and G', N', M' the isogonals of G, N, M. Call <P> the...
... GENERALIZATION (conjecture): If two orthologic triangles are perspective, then the perspector is collinear with the two orthologic centers! NOTE: Two...
The following paper has been published in Forum Geometricorum; it can be viewed at http://forumgeom.fau.edu/FG2003volume3/FG200305index.html The Editors, Forum...
ForumGeom
ForumGeom@...
Mar 3, 2003 3:11 pm
6637
In response to Darij's inquiry - The Yff hyperbola was introduced by Peter Yff in an undated paper in Annals of the New York Academy of Sciences, vol. 500,...
Dear Jean-Pierre, ... [JP] ... the reflections of the lines AA', AB', AC' about a bisector at A in ABC intersect the sidelines of A'B'C' at three collinear...
Dear Clark Kimberling and Bernard Gibert, Thank you very much. By the way, Peter Yff had written in a private ... So the paper may be dated, though. But let me...
Dear Bernard ... M,N ... [Bernard] ... in ABC ... triangle ... in ABC and ... Many thanks and congratulations for this wonderful and so clever construction. ...
Dear Hyacinthians, If P = x:y:z (trilinears), then W(P) denotes (TCCT p 238) the following inscribed conic: (x^2)(alpha)^2 + (y^2)(beta)^2 + (z^2)(gamma)^2 -...
Dear Clark Kimberling, ... [...] ... I. e. you state that P lies on W(P) if and only if the trilinear polar of P passes through the incenter. Here I am not...
There are 32 sets of three Malfatti circles. Is it two of these 32 that is referred to ? R....
Richard Guy
rkg@...
Mar 3, 2003 8:21 pm
6645
... I found a completely different proof of that result. We are talking about synthetic proofs, since the result is trivial computationally. Lemma: If...
Dear Barry Wolk, Many thanks for the proof. It is indeed simpler. By the way, the Corollary is equivalent to the theorem on Orthologic Triangles, isn't it? ...
Dear Clark, ... your W(P) is in fact the in-conic whose perspector is the isogonal conjugate P* of P. Its center Q is therefore the complement of the isotomic...
Dear Richard Guy, ... Unfortunately, I have not seen the 32 sets. If they are all possible sets of Malfatti-touching circles, then the Malfatti circles and the...
... ^^^^^^^^^^ correctly: Longchamps point ... Now I found that the author of the problem was J. W. Clawson, and thus we have two Clawson points, X(19) and...
... By the way, the point Tb has trilinear coordinates ( csc(C-B) : csc B : csc(A-B) ). ... [...] ... In fact, they don't concur. Triangle TaTbTc is not in...
Darij has recently discussed an Arnold theorem (for a given point P the perpendicular from A to HPa meets BC at X, etc, then X,Y,Z are collinear). I am left...
... Orthologic ... [snip] ... is ... then ... Yes it is equivalent. I untangled the notation last night. Orthologic theorem : If perpendiculars from A to B'C',...
Dear Dick Tahta, Barry Wolk, Hyacinthos members, Many thanks for the notes on synthetic and analytic proofs. At first, many of you already know that I try to...
Dear Barry Wolk, Hyacinthos members, ... Thank you for verification. Let me describe the context of the conjecture and my search for a further generalization...
Dear friends, Regard the isogonal conjugates of all points on a circle k with respect to triangle ABC. They seem to lie on a tricusp hypo/hypercycloid(?). (1)...
Dear friends, This is a followup to the problem of messages #6566, #6575, #6576: The circle DF(1,1) is orthogonal to the polar circle of triangle ABC. (Proof...
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