I think I have posted some of this already way back, but it's not trivial to browse the archive. :-) Basic construction: Let ABC be a triangle and D a point in...
To recap: Build a tetrahedron ABCD and project D into the plane ABC to get P. Demand f(AB,CD)=f(AC,BD)=f(BC,AD). Let f be... P traces... quotient -> Euler...
One free book I like (from http://pdg.lbl.gov/ ) is the Particle Data Booklet. Anyone can get it free, and it is updated roughly yearly. One new section is...
Dear Hauke [HR] ... For the sum and the difference the locus is the same, The Soddy line passing through the ETC points 1 7 20 77 170 175 176 269 279 347 390 ...
... Are you sure? With MATHEMATICA, I get the following. Place Axyz,Bxyz,Cxyz at standard Kimberling values {-13/3, (4*Sqrt[35])/3, 0, 6, 0, 0, 0, 0, 0} and D...
Dear Hauke. I think that locus is part of the Soddy line. Not all the points have the property of the locus. The same holds for the locus of sum. For example...
... Aaargh, spurious roots again! My "special point" was #3160 (=(#175+#176)/2 which makes perfect sense) and when I plug in the "solution", BC+AD=AC+BD=AB+CD...
Dunno if Darij already reported this (when we produced radical centers by the score, the ETC was *much* shorter). Anyway. Draw three circles around A,B,C with...
1. Draw circles with radii b,c,a around A,B,C. Call the radical center of these circles R1. 2. Draw circles with radii c,a,b around A,B,C. Call the radical...
Which of these conditions guarantee that a triangle is isosceles? a) Two altitudes have same length. b) Two side bisectors have same length. c) Two angle...
The concept of bisector of a side is not clear...the perpendicular bisector is a straight line, and not a segment. With respect to the angle bisectors, the...
Dear Hauke, ... I think as side bisectors you mean ABC medians and a), b) guarantee that ABC is isosceles but c) guarantee that ABC is isosceles only if we...
... Can you give a "pupil-niveau" proof of c)? a) and b) were trivial one-liners even for me :-) I was a bit paranoid that one of the cases would turn out to...
Of course I'm only joking - what I mean is that I work purely algebraically and thus can use wacky things like circle with negative or even imaginary radii...
Here is a table with radical centers (I probably already posted it, but since ETC is a bit larger now, I could amend one - gasp! :-) - entry.) Draw circles...
Dear Hyacinthists, Problem 13 from the link below: TC given (H_a,I,I_a). I would like to have a hint to the TConstruction. Let D be the midpoint of the segment...
Dear Luis, If the line II_a meets BC at E then since BI, BI_a are bisectors of <ABC the points A, E are the harmonic conjugates relative to I, I_a. Since AH_a...
Dear Hauke, ... I think that the most simple proof of c) is the following: If the internal angle bisectors BD, CE of triangle ABC are equal and AC > AB then <B...
Dear Hyacinthists, Nikos Dergiades and Alexei Myakishev, [AM] But don't please publish the solution until April, 1.[AM] Sincerely AlexeyOk. After having sent...
Someone always had had the idea before, but anyway :-) Consider a center foo of triangle ABC to have trilinears x:y:z. Then I define the outer foo centers to...
Oh, I see, the formatting comes only good viewed in reply mode. ... I should add that from power line magic it also follows that all point with constant j lie...
I think will be useful for the organizers of Contests, Olympiads, and editors of journals with problems that when this type of announcement be made, it...
Dear Hauke, ... Your triangle XYZ is the anticevian triangle of the circumcenter O. Are you sure that X(6) of XYZ is X(6) of ABC? I think that is very...
Dear Francisco, ... You are right. When I answered to the message of LuĂs Lopes I did n't thought that the ending date of sending solutions had not passed....
Dear Hauke, [HR] ... Yahoo suppresses "white space" in our messages. We can view the format correctly by first selecting 'Show Message Option' followed by 'Use...
... So that generalizes? What I called "outer centers" of foo is in general anticevian triangle w.r.t. center foo? (I'm not good at interpreting trilinears!) ...
Dear Hauke, I don't use Mathematica. With sketchpad construct a triangle ABC, the triangle XYZ and the points P = X(6) of XYZ and Q = X(6) of ABC and see if P...
Dear Hyacinthians, I am struggling with the following problem: Version A --- Let be given a quadrilateral ABCD (which may be non-convex or self-intersecting)...
Dear Eisso, I have found the following: If ABC is a triangle, the locus of D so that E, F, G and M are concyclic is a sextic passing through A, B, C and the...
Dear Hyacinthists, e_b := B external bisector One can construct (Euclides) 2 triangles given (A,e_b,r_a). But how about (A,e_b,r_c)? I wasn't able to construct...
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