Thanks!!!!! I would like the n-th ratio be something like this: [f(n)/g(n)] * F(a,b,c) with f(n), g(n) integer sequences, which I would send to seqfan...
Dear geometers: May anyone, please, confirm the coordinates of X(5448) in ETC? I get barycentrics ...
21955
Francisco Javier
garciacapitan
Apr 15, 2013 7:21 am
You're right....
21956
César Lozada
cesar_e_lozada
Apr 15, 2013 10:28 am
Dear geometers: If P=(p : q : r ) and U=( u : v : w), what’s the geometrical meaning of Q = ( p*u : q*v : r*w) for barycentric and trilinear coordinates, if...
21957
Antreas Hatzipolakis
xpolakis
Apr 15, 2013 6:37 pm
I do not find a method to construct P*U, but there are simple loci (ie we can find them by simple computations) ... line? aph ... -- [Non-text portions of this...
21958
Bernard Gibert
bernardgibert
Apr 15, 2013 6:46 pm
Dear César Lozada, ... You may want to consult §1.2.2 in http://bernard.gibert.pagesperso-orange.fr/files/isocubics.html Best regards Bernard [Non-text...
21959
Barry Wolk
wolkbarry
Apr 15, 2013 6:47 pm
... This was already answered, and is the whole plane. Another way of stating this result is: The Poncelet points of the quads BCPP*, CAPP*, ABPP* form a...
21960
Antreas
xpolakis
Apr 15, 2013 6:50 pm
I guess that for P = I (instead of N) the ratio should be simpler.... ie Let A1, B1, C1 be the NPC centers of IBC, ICA, IAB and O1 the circumcenter of A1B1C1. ...
21962
Francisco Javier
garciacapitan
Apr 15, 2013 7:35 pm
[APH] Which is the locus of P such that the circumcenter of A1B1C1 is on the Euler line (where A1,B1,C1 are the NPC centers of PBC, PCA, PAB)? ... It is a...
21963
rhutson2
Apr 15, 2013 9:24 pm
For P = O, Op = complement of X(157). In general, Op = complement of the nine-point-center of the antipedal triangle of P. Randy...
21964
Antreas Hatzipolakis
xpolakis
Apr 16, 2013 9:29 am
... Euler line (where A1,B1,C1 are the NPC centers of PBC, PCA, PAB)? ... I think it is the same with Bernard's Q038: ...
21965
forumgeom forumgeom
ForumGeom@...
Apr 16, 2013 3:32 pm
The following paper has been published in Forum Geometricorum. It can be viewed at http://forumgeom.fau.edu/FG2013volume13/FG201309ndex.html The editors Forum...
21966
rhutson2
Apr 16, 2013 10:17 pm
Dear friends, Let ABC be a triangle, and P a point. Let A'B'C' be the pedal triangle of P. Let Ba, Ca be the orthogonal projections of A' onto lines CA, AB,...
21967
Antreas
xpolakis
Apr 16, 2013 10:30 pm
Let ABC be a triangle. A conic intersects sidelines BC at A1, A2, CA at B1, B2 and AB at C1, C2. Theorem: (BA1 / CA1) * (BA2 / CA2) * (CB1 / AB1) * (CB2 / AB2)...
21968
Antreas
xpolakis
Apr 16, 2013 10:43 pm
Let ABC be a triangle and P a point. Which is the locus of P such that the NPC center of PBC lies on the line AP ? Ceva triangle variation: Let ABC be a...
21969
Bernard Gibert
bernardgibert
Apr 17, 2013 7:46 am
Dear Randy, ... a quintic with many simple points but only two (I think) ETC centers : X4, X1498. ... seems very difficult... Best regards Bernard [Non-text...
21970
Antreas Hatzipolakis
xpolakis
Apr 17, 2013 10:03 am
A CIRCLE: Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3 the NPC centers of IB'C', IC'A', IA'B', resp. The points I, N1,N2,N3 are...
21971
César Lozada
cesar_e_lozada
Apr 17, 2013 1:04 pm
Yes, they are concyclic. The center X of the circle has trilinears: 2*cos(A)+4*sin(3*A/2)*cos(B/2-C/2)+ cos(B-C)+2 : : ETC search: 2.387773069046934..,...
21972
César Lozada
cesar_e_lozada
Apr 17, 2013 2:45 pm
Correction: X lies on line X(I),X(J) for these (I,J): (1,30), (3,81), (5,581), (21,323), (58,5428), (140,3216), (186,2906), (386,549), (511,1385), (550,991),...
21973
yiuatfauedu
Apr 17, 2013 2:58 pm
Dear Randy and Bernard, [RH] Let ABC be a triangle, and P a point. Let A'B'C' be the pedal triangle of P. Let Ba, Ca be the orthogonal projections of A' onto...
21974
Antreas
xpolakis
Apr 17, 2013 6:16 pm
... The most interesting line is the first one (X1,X30): If I0 is the center of the circle, then the line II0 is parallel to Euler line of ABC. There are three...
21975
rhutson2
Apr 17, 2013 7:37 pm
Paul, This is interesting: the perspector you mention with ETC search value 5.10435062529 matches the isogonal conjugate of the polar conjugate of X(1073), and...
21976
Paul Yiu
yiuatfauedu
Apr 17, 2013 8:18 pm
Dear Randy, Yes, S_{AB} is shorthand for (S_A)(S_B) etc. I always use barycentric coordinates unless the context clearly favors trilinear coordinates. Let's...
21977
rhutson2
Apr 18, 2013 1:33 am
A related locus: Q such that Q and the NPCs of BCQ, CAQ, ABQ are concyclic. This would include X(13), X(14), the bicentric pair PU(5), the circumcircle...
21980
Antreas
xpolakis
Apr 18, 2013 6:11 am
Let ABC be a triangle and P a point. Denote: L1, L2, L3 = the Euler Lines of PBC, PCA, PAB, resp. R1 = the radical axis of ((NPC_PBC), (O_PBC)) ie the radical...
21981
Francisco Javier
garciacapitan
Apr 18, 2013 6:30 am
This is the tricircular sextic Q014 - 4 S^2 x y z (x + y + z) (c^2 x y + b^2 x z + a^2 y z) where S=twice the area of ABC. (tricircular = the circular points...
21982
rhutson2
Apr 18, 2013 8:20 am
Francisco Javier, While Q014 mentions PU(5), it does not contain them, nor X(13) and X(14), which are on this curve. Also, X(80) which is on Q014, is not on...
21983
rhutson2
Apr 18, 2013 10:58 am
My apologies. I misread Francisco Javier's post below. As he explained privately, he meant the difference: LOCUS = SEXTIC(Q014) - SEXTIC(S 4 S^2 x y z (x + y...
21985
Antreas
xpolakis
Apr 18, 2013 1:15 pm
Let ABC be a triangle, P a point, and (Op) the cevian circle of P (with center Op). Which is the locus of P such that the Poncelet point of Op (ie the point of...
21986
xpolakis
Apr 19, 2013 6:06 am
1. Hyacinthos is closed. The archive is available. 2. Hyacinthos members may subscribe to ANOPOLIS list: Related Link: http://groups.yahoo.com/group/Hyacinthos...
