Dear Jean-Pierre Thanks [APH] ... Variation: Lab := The Reflection of BB' in AA' Mab := The Parallel to Lab through B (instead of C). Lac := The Reflection of...
Dear Hyacinthists, First of all, thank you to Peter and Nikos for their help with my last question. Here CD_c = d_c internal C bisector. There are many known...
Let ABC be a triangle, L a line = the Euler line for example, and P a point on L. The Circumcircles of PBC, PCA, PAB intersect again the line L at A',B',C',...
Dear Jean-Pierre [APH] ... [JPE] ... Another problem is to find the point(s) P on the given line L, such that the Euler lines of AB'C', BC'A', CA'B' are...
[APH] ... [JPE] ... EQUIVALENTLY: Let ABC be a triangle, A1B1C1 the circumcevian triangle of H, and M1M2M3 the Medial triangle of ABC. Let A*,B*,C* be the...
... [JPE] ... From this we can make this generalization: Let ABC be a triangle, A1B1C1 the circumcevian triangle of point P, and M1M2M3 the Medial triangle of...
Let ABC be a triangle P, a point and A*B*C* the reflection of ABC in P (ie A* = the reflection of A in P, etc) I think that the circumcircles of (1) A*BC,...
... Dear Nikos as parome ta pramata apo thn arxh: Loipon, exome kai leme: 1. Let ABC be a triangle, O its circumcenter, and A*B*C* the circumcevian triangle of...
... A circumconic with center P = [x:y:z] a for barycentic equation x(y+z-x)vw + y(z+x-y)wu +z(x+y-z)uv = 0 Or x(y+z-x)/u + y(z+x-y)/v +z(x+y-z)/w = 0 For...
Dear Michael, Antreas wrote ... I answered, ... and you wrote ... I didn't understand. You mean that the denominator a gave (y+z-x)(yc^2+zb^2)-2yza^2 is not...
Let ABC be a triangle, and P,P* two isogonal conjugate points. Denote: A',B',C' := The reflections of A,B,C in P, resp. A",B",C" := The reflections of A,B,C in...
Let ABC be a triangle, P a point,and Na, Nb, Nc the NPC centers of PBC,PCA,PAB, resp. Which is the locus of P such that ABC, NaNbNc are orthologic? I think...
Dear Antreas, wellcome from Crete. ... If Pa is the projection of P on BC, A' is the midpoint of BC projection of O on BC then since the NPC of PBC passes...
Dear Nikos [ND] <--- epikairo akronymio! ... Thanks [APH] ... [ND] ... So the one orthologic center [NaNbNc, ABC] is a homothetic image of P with center O. I...
Dear Antreas, ... Pio epikairo einai to Niko LA.O.S Eviva Rethymno kai prwtistws (Arkadiou,Koukoulwna,Kouritwn,Lappaiwn,Nikiforou,Sybritou) ... I don't know. I...
Dear Antreas and Nikolaos, ... the second orthologic center is the antigonal of P. http://pagesperso-orange.fr/bernard.gibert/gloss/pointsandmapping.html Best...
APH said (rephrased): Let ABC be a triangle and P a point. Let A1,B1,C1 be the NPC centers of PBC,PCA,PAB, resp. Which is the locus of P such that ABC, A1B1C1...
Let ABC be a triangle and A1,B1,C1 the NPC centers of NBC,NCA,NAB, resp. Which point is the Circumcenter O1 of the triangle A1B1C1? (on the Euler Line of ABC...
... O1 is on the Euler line of ABC but O2 does not. The barycentric coordinates of O1 are: {2 a^16 - 9 a^14 b^2 + 13 a^12 b^4 + a^10 b^6 - 25 a^8 b^8 + 33 a^6...
Dear Francisco Thank You! I do not know if the point meets Clark's criteria, in order to be included in ETC (new points appeared recently in Hyacinthos did not...
[APH] ... [FGC] ... [snip] A generalization: Let ABC be a triangle, and P a point on the Euler Line of ABC. Let Pa, Pb, Pc be the P-points wrt triangles PBC,...
Dear Hyacinthists, In Angel Montesdeoca Delgado's web page below one finds an interesting solution to the problem (a,h_a,r). In it there is the conic (actually...
Dear Luis, If I understood what you want, then y_0 = a^2r/(a^2-4r^2) u = 4r^3/(a^2-4r^2) v = 2r^2/sqrt(a^2-4r^2) where a^2 > 4r^2 always and the focii are ( 0...
Dear Antreas For P = O you are right, but not all points on the Euler line have this property. For example take as P, X(405) the intersection of Euler line and...
Dear Nikos Dergiades, Thank you very much. Garcia Capitan had already given me the same results privately. === and the focii are ( 0 , 2ar^2/(a^2-4r^2) ) ( 0...
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