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...
17830
LuÃs Lopes
qedtexte
Jun 1, 2009 1:41 pm
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...
17831
xpolakis
Jun 1, 2009 8:19 pm
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',...
17832
jpehrmfr
Jun 1, 2009 9:29 pm
Dear Antreas ... Your point P* lies upon the circumcircle of ABC (for any line L) Jean-Pierre...
17833
xpolakis
Jun 1, 2009 10:17 pm
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...
17837
xpolakis
Jun 3, 2009 8:52 pm
... [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...
17838
xpolakis
Jun 3, 2009 10:46 pm
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,...
17839
Nikolaos Dergiades
ndergiades
Jun 4, 2009 6:49 am
Dear Antreas, ... Yes. The common point has first barycentric (y+z-x)/[(y+z-x)(yc^2+zb^2)-2yza^2] where P = (x:y:z) Best regards Nikos ...
17841
xpolakis
Jun 4, 2009 11:50 am
... 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...
17842
michgarl
Jun 6, 2009 12:39 pm
... Dear Nikos and Antreas When you develop the denominator of the common point it is " c²*y² + b²*z² +2*y*z*S_a - a²*y*z-b²*z*x-c²x*y " First...
17843
michgarl
Jun 8, 2009 6:27 am
... 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...
17844
Nikolaos Dergiades
ndergiades
Jun 8, 2009 8:27 am
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...
17845
xpolakis
Jun 8, 2009 11:45 am
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...
17846
xpolakis
Jun 8, 2009 1:52 pm
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...
17847
Nikolaos Dergiades
ndergiades
Jun 8, 2009 2:46 pm
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...
17848
xpolakis
Jun 8, 2009 10:39 pm
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...
17849
Nikolaos Dergiades
ndergiades
Jun 9, 2009 11:08 am
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...
17850
Bernard Gibert
bernardgibert
Jun 9, 2009 12:40 pm
Dear Antreas and Nikolaos, ... the second orthologic center is the antigonal of P. http://pagesperso-orange.fr/bernard.gibert/gloss/pointsandmapping.html Best...
17851
xpolakis
Jun 9, 2009 4:48 pm
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...
17852
xpolakis
Jun 9, 2009 10:07 pm
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...
17853
xpolakis
Jun 11, 2009 5:40 pm
[APH] ... Is this point on the Euler Line of ABC? Coordinates? APH...
17854
garciacapitan
Jun 11, 2009 6:36 pm
... 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...
17855
xpolakis
Jun 11, 2009 6:50 pm
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...
17856
xpolakis
Jun 11, 2009 7:48 pm
[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,...
17857
Luís Lopes
qedtexte
Jun 12, 2009 1:05 pm
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...
17858
Nikolaos Dergiades
ndergiades
Jun 12, 2009 3:32 pm
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...
17859
Nikolaos Dergiades
ndergiades
Jun 12, 2009 4:05 pm
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...
17860
LuÃs Lopes
qedtexte
Jun 12, 2009 4:11 pm
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...
