Anopolis Yahoo Group CONCURRENT NPCs =============================================== xpolakis Message 1 of 5 May 23, 2009 ----------------------------------------------- Let A,B,C,D be four points (not an orthocentric tetrad). Let (1),(2),(3),(4) be the NPCs of the triangles BCD,CDA,DAB,ABC, resp. concurrent at P. Denote: Mij = The other than P intersection of the cirles (i) and (j). Mi*j = The antipode of Mij in the circle (i) Mij* = The antipode of Mij in the circle (j) (ij) = The NPC of MijMi*jMij* The circle (ij) pass through P. Now there are 10 concurrent circles at P [we have (ij) = (ji)]: (1),(2),(3),(4), (12), (13),(14), (23), (24), (34) Next, we have 49 circles: (1),(2),(3),(4), (12), (13),(14), (23), (24), (34), (1(12)), (1(13)), (1(14)), (1(23)), (1(24)), (1(34), (2(12)), (2(13)), (2(14)), (2(23)), (2(24)), (2(34)), (3(12)), (3(13)), (3(14)), (3(23)), (3(24)), (3(34)), (4(12)), (4(13)), (4(14)), (4(23)), (4(24)), (4(34)), ((12)(13)), ((12)(14)),((12)(23)), ((12)(24)), ((12)(34)) ((13)(14)),((13)(23)), ((13)(24)), ((13)(34)), ((14)(23)), ((14)(24)), ((14)(34)), ((23)(24)), ((23)(34)), ((24)(34)) Next, we have 1180 circles. So we have the sequence: a_1 = 4 a_2 = 10 a_3 = 49 a_4 = 1180 ... And in general : a_1 = 4 a_n+1 = 4 + C(a_n, 2) Antreas =============================================== Antreas Message 2 of 5 Jul 17, 2013 ----------------------------------------------- Listed in OEIS http://oeis.org/A151611 aph =============================================== Antreas Hatzipolakis Message 3 of 5 Aug 26, 2016 ----------------------------------------------- 1. Orthic triangle version: Let ABC be a triangle and A'B'C' the orthic triangle. Denote: JaaJabJac = the excetral triangle of AB'C' (the excircle (Jaa) touches B'C', the excircle (Jab) touches AC', the excircle (Jac) touches AB') JbaJbbJbc = the excentral triangle of BC'A' (the excircle (Jba) touches BC', the excircle (Jbb) touches C'A', the excircle (Jbc) touches BA') JcaJcbJcc = the excentral triangle of CA'B' (the excircle (Jca) touches CB', the excircle (Jcb) touches CA', the excircle (Jcc) touches A'B') The NPCs of A'JabJac, B'JbcJba, C'JcaJcb are concurrent. 2. Excentral triangle version: Let ABC be a triangle and IaIbIc the excentral triangle. Denote: IaaIabIac = the excentral triangle of IaBC (the excircle (Iaa) touches BC, the excircle (Iab) touches IaC, the excircle (Iac) touches IaB) IbaIbbIbc = the excentral triangle of IbCA (the excircle (Iba) touches IbC, the excircle (Ibb) touches CA, the excircle (Ibc) touches IbA) IcaIcbIcc = the excentra triangle of IcAB (the excircle (Ica) touches IcB, the excircle (Icb) touches IcA, the excircle (Icc) touches AB) The NPCs of AIabIac, BIbcIba, CIcaIcb are concurrent Antreas =============================================== Antreas Hatzipolakis Message 4 of 5 Aug 26, 2016 ----------------------------------------------- [APH]: 1. Orthic triangle version: Let ABC be a triangle and A'B'C' the orthic triangle. Denote: JaaJabJac = the excetral triangle of AB'C' (the excircle (Jaa) touches B'C', the excircle (Jab) touches AC', theexcircle (Jac) touches AB')  JbaJbbJbc = the excentral triangle of BC'A' (the excircle (Jba) touches BC', the excircle (Jbb) touches C'A', theexcircle (Jbc) touches BA') JcaJcbJcc = the excentral triangle of CA'B' (the excircle (Jca) touches CB', the excircle (Jcb) touches CA', theexcircle (Jcc) touches A'B') The NPCs of A'JabJac, B'JbcJba, C'JcaJcb are concurrent.  [Peter Moses]: Hi Antreas, 1) X(389) Best regardsPeter. =============================================== Antreas Hatzipolakis Message 5 of 5 Aug 26, 2016 -----------------------------------------------  [APH]:2. Excentral triangle version: Let ABC be a triangle and IaIbIc the excentraltriangle. Denote: IaaIabIac = the excentral triangleof IaBC (the excircle (Iaa) touches BC, the excircle (Iab) touches IaC,the excircle (Iac) touches IaB) IbaIbbIbc = the excentral triangleof IbCA (the excircle (Iba) touches IbC, the excircle (Ibb) touches CA,the excircle (Ibc) touches IbA) IcaIcbIcc = the excentra triangleof IcAB (the excircle (Ica) touches IcB, the excircle (Icb) touches IcA, theexcircle (Icc) touches AB) The NPCs of AIabIac, BIbcIba, CIcaIcbare concurrent [Peter Moses]: Hi Antreas, 2) X(10). Best regards,Peter. =============================================== Cached by Georg Fischer at Nov 14 2019 12:46 with clean_yahoo.pl V1.4