Dear Barry Wolk, ... Another high school student (me) is giving a solution he has found right off. I guess it can be simplified a lot. I will use the POMPEIU...
Dear Georg and Jean-Pierre, ... The solution I wrote this morning assumed that A'AG shouls be collinear. That is a wrong assumption, so basicly all I wrote was...
Dear Milorad Stevanovic, ... This is X(177), the 1st Mid-Arc point. Indeed, the barycentrics given in Kimberling's ETC: ( (sin A)(cos B/2 + cos C/2) sec A/2 :...
Dear friends, I cite Kimberling's ETC (with slight modification): Let A', B', C' be the first points of intersection of the angle bisectors of triangle ABC...
Regard a triangle ABC and a point P. If x1, y1, z1 are the distances from P to the sidelines BC, CA, AB (and x1 > 0 for P on the same half-plane of BC as A,...
If A'B'C' is the Yff central triangle of a triangle ABC, then A' has trilinear coordinates ( cos(B/2) cos(C/2) ... On the other hand, let I be the center of...
Dear Eric, [ED]: I call the triangle formed by the points F'a, F'b and F'c where the ... I have just confirmed your result. Indeed the same is true for the...
Given a triangle ABC. We have a triangle XYZ which is perspective to all cevian triangles of triangle ABC. Is XYZ necessarily an anticevian triangle? If yes: ...
Dear Paul Yiu, ... Very nice observation, but "happens" sounds as if it would be some unexpected coincidence. In fact, there is a simple reason: After the...
... [PY]: I have just confirmed your result. Indeed the same is true for the cevian ... It is quite interesting to note that the same is true also for X(970),...
Dear Eric and Paul ... There should be a locus of points with that property. Which is it? ... ... and somebody else wrote once something like that, but about...
Dear Darij ... Consider a non-degenerated triangle U,V,W and suppose that U,V,W don't lie on the sidelines of ABC. In any system of projective coordinates with...
Dear Paul and Antreas ... [APH] ... Do you mean that we have to wait about three centuries a new Andrew Wiles to know these coordinates? Friendly. Jean-Pierre...
Dear Antreas and Jean-Pierre, [PY]: But the coordinates of the perspector is too complicated to write down here. [APH] ... and somebody else wrote once...
Dear Darij ... Obviously, these cubics are exactly the cubics with the sidelines of ABC as "flex asymptots" - sorry, but I don't know the English word: I mean...
Dear Paul ... new ... double ... are. I'm very sorry for the time you've lost in this tedious computation. Of course, following Antreas, I was trying a very...
Dear Jean-Pierre Ehrmann, ... Many thanks for the solution! I was stupid as I thought there should be a finite number of cevian triangles. Sincerely, Darij...
Dear friends, Let A1,B1,C1 be the second intersections of AI,BI,CI with three excircles. Then 1.Ttriangles A1B1C1 and medial triangle of triangle ABC are...
Dear friends, Given a segment AB of length c, and another segment of length ell, construct in an elegant way a triangle ABC with a + b = ell and a^3 + b^3 =...
... Dear Paul, I don't see what property has the special triangle with a^3 + b^3 = c^3 (to apply it for an elegant solution), but in general: if a + b = l, and...
From an old book (from the old good days, when geometry was .... geometry!): Construct a triangle if are given: a, h_a [=altitude from A], 3b^2 + 2c^2 = min. ...
Dear friends, Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. Then the centroid G'of triangle A'B'C' has...
Dear Antreas, If we denote by D the feet of perpendicular from A to BC and take BD=a-x,DC=x then 3(b^2)+2(c^2)=7(x^2)-4ax+2(a^2)+3(h_a)^2 has minimum for...
I think I have already studied partly the following configuration in an old thread. Let ABC be a triangle, P a point, and PaPbPc its pedal triangle. The line...
Let ABC be a triangle, P a point, and PaPbPc its pedal triangle. The line PPa intersects AB at Ab, and Ac at Ac. Let Mac = midpoint of AcP Mab = midpoint of...
... Dear all, I would like to know the date of birth and the date death of Antoine Gob He was a teacher in Hasselt (Belgium) and published with J. Neuberg a...
Dear Milorad ... How did you obtain that equation? I think that if we apply twice the Pythagorean theorem (in the triangles DAB, DAC) we get: 3b^2 + 2c^2 =...
