... ^^^^^^^^^ JPE says "of course" because the isogonal points P, gP share the same pedal circle (therefore both P, gP are points of the locus). But how about...
2918
John Conway
conway@...
May 28, 2001 5:58 pm
... Let me briefly comment on the use of the terms "isogonal"; and "self-isogonal" applied to curves. It is NOT a good idea to make these mean the same thing, ...
2919
Bernard Gibert
b.gibert@...
May 28, 2001 7:14 pm
Dear friends, I know terminology is a very important matter in any kind of human activity, and of course, in geometry. I think good terminology should be, AS...
2920
John Conway
conway@...
May 28, 2001 7:42 pm
... [then lots of words I agree with, followed by:] ... Yes, this certainly arose as the result of a misunderstanding on my part, but it doesn't matter one...
2921
xpolakis@...
May 29, 2001 5:53 pm
Let ABC be a triangle and P a point (on the plane of the triangle). The line through P and parallel to BC intersects AB at Ab and AC at Ac CA BC...
2922
xpolakis@...
May 29, 2001 5:54 pm
Let ABC be a triangle and AD the altitude from A. DC AC If BC is fixed, and -- = (--)^2, which is the locus of A ? DB AB APH...
2923
yiu@...
May 29, 2001 6:15 pm
Dear Antreas, [APH]: Let ABC be a triangle and P a point (on the plane of the triangle). The line through P and parallel to BC intersects AB at Ab and AC at Ac...
2924
yiu@...
May 29, 2001 6:19 pm
Dear Antreas, [APH]:Let ABC be a triangle and AD the altitude from A. DC AC If BC is fixed, and -- = (--)^2, which is the locus of A ? DB AB ... The...
2925
xpolakis@...
May 29, 2001 6:33 pm
Dear Paul, ... Dear Paul, That's at first glance. At second glance, the locus is something more: (circle with diam BC) + (perp. bis. of BC) and at third...
2926
yiu@...
May 29, 2001 6:45 pm
Dear Antreas, ... converse ... That is what you call an ``impure'' locus. ... And the rectangular hyperbola with B and C as vertices! Best regards Sincerely ...
2927
John Conway
conway@...
May 29, 2001 6:47 pm
... What's the origin of this name? Let me ask another question. For what points are the distances from A_P to A_[P], B_P to B_[P], C_P to C_[P] equal?...
2928
xpolakis@...
May 29, 2001 6:57 pm
... Dear Paul, I think that this case is but our "shadows" case, except that instead of sines in shadows we have here cscines. That is, the shadow of the side...
2929
Paul Yiu
yiu@...
May 29, 2001 7:11 pm
Dear John, ... I found this in [TCCT], where Clark called the point X(182) ``equal parallelians point''. It appeared in Vigarie's ``19th century of ...
2930
yiu@...
May 29, 2001 7:17 pm
Dear Antreas, If p:q:r = a:b:c, then of course we get (... : 1/c^2 + 1/a^2 - 1/b^2 : ...) What does John call this? On the other hand, if these parallelians...
2931
xpolakis@...
May 29, 2001 7:30 pm
Dear Paul, ... I think that we have to investigate it systematically in an follow-up of our paper, where also we may investigate the cases of shadows of the...
2932
John Conway
conway@...
May 29, 2001 7:34 pm
... "the dilated quocentroid" dqG, or just the "dilated Quotient" point Q. For recall that the "quotient"; map takes <:Y:> -> <:Y/bb:> and the dilation one...
2933
John Conway
conway@...
May 29, 2001 7:36 pm
... Thanks Paul - unfortunately it seems that we must go to the references to see why he named it so!? JHC...
2934
Paul Yiu
yiu@...
May 29, 2001 7:45 pm
Dear John, ... It is the superior [dilation?] of the isotomic conjugate of the Mittenpunkt. Too long? Best regards Sincerely Paul...
2935
Floor van Lamoen
f.v.lamoen@...
May 29, 2001 7:45 pm
... Isn't it just the point through which pass three "parallel intercepts" of equal lenghts (a-parallel intercept: parallel to a, intercepted by b and c)? Kind...
2936
xpolakis@...
May 29, 2001 8:27 pm
... It just occured to me that we can generalize the problem in this way: The segment AbAc is parallel to BC, therefore it is perp. to altitude from A, and...
2937
John Conway
conway@...
May 29, 2001 9:16 pm
... Let me check this! The ends of the b-parallel through <X:Y:Z> are <X+Z,Y,0> and <0,Y,X+Z>, whose difference is X+Z times b. So, yes, we want the...
2938
John Conway
conway@...
May 29, 2001 9:19 pm
... In symbols drMo, or dtMo in an older notation we might revert to "the direciprocal Mittenpunkt", or "di-(iso)tomic Mittenpunkt". Not too long! JHC...
2939
xpolakis@...
May 29, 2001 10:09 pm
Let ABC be a triangle and Na, Nb, Nc the NPC centers of the triangles IBC, ICA, IAB [I = Incenter of ABC]. Prove that the triangles: medial (of ABC) and NaNbNc...
2940
Steve Sigur
ssigur@...
May 30, 2001 1:18 am
... John, Kapetis has a very nice section about this. Consider a point P in the triangle plane and draw parallels to the sides through it. There is a point ...
2941
John Conway
conway@...
May 30, 2001 1:43 pm
... Thanks, Steve. It is in fact the di-quo-incenter, as was made clear in the message I was responding to, and Floor had already told me this definition;...
2942
xpolakis@...
May 30, 2001 3:52 pm
Another generalization: Let ABC be a triangle and A'B'C' the pedal (or cevian) triangle of a fixed point Q. Let P be a variable point, and: The line through P...
2943
xpolakis@...
May 30, 2001 4:56 pm
As the center of an inconic moves on a fixed line (the Euler, for example), where are its foci moving on? This is an old problem I don't remember where I read...
2944
Floor van Lamoen
f.v.lamoen@...
May 30, 2001 6:29 pm
... Luckily Clark Kimberling doesn't use this term. He calls the point "equal parallelians point". And parallelians are what I called parallel intercepts...
2945
xpolakis@...
May 31, 2001 5:27 pm
Consider the following schema: A /\ / \ Ca Ba / \ / \ Cb P Bc / \ / Ha \ B---Ab-------Ac--C Denote ang(PAbAc) := Ab,...
2946
yiu@...
Jun 1, 2001 1:35 am
If we consider the Kenmotu configuration. Draw lines through the vertices of the three squares, nearer to the respective vertices of the triangle, parallel to...
