http://tech.groups.yahoo.com/group/Anopolis/ Anopolis Wed, 29 Jul 2009 23:05:02 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/134 http://tech.groups.yahoo.com/group/Anopolis/message/134 GENERALIZATION: Let ABC be a triangle, and Pa,Pb,Pc three perpendiculars to BC,CA,AB, resp. Let L be a line, and Ra,Rb,Rc the reflections of Pa,Pb,Pc in L, Tue, 28 Jul 2009 17:49:09 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/133 http://tech.groups.yahoo.com/group/Anopolis/message/133 Let ABC be a triangle, P point and P* its isogonal conjugate. Let A'B'C', A"B"C" be the pedal triangles of P,P*, resp. Denote: L := The a line passing through Sun, 19 Jul 2009 16:50:04 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/132 http://tech.groups.yahoo.com/group/Anopolis/message/132 ... GENERAL THEOREM (Proof?): Let ABC be a triangle and P, Q two points. Denote: ApBpCp := The CEVIAN triangle of P AqBqCq := The PEDAL triangle of Q Hp, Hq := Fri, 17 Jul 2009 20:51:56 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/131 http://tech.groups.yahoo.com/group/Anopolis/message/131 Theorem (proof?) : Let A'B'C' be the cevian triangle of I wrt. ABC and let H' be the orthocenter of A'B'C'. Then, the reflections of IH' wrt. the sidelines Tue, 14 Jul 2009 21:09:23 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/129 http://tech.groups.yahoo.com/group/Anopolis/message/129 1. Let ABC be a triangle and A1,B1,C1 the circumcenters of NBC,NCA,NAB, resp. [where N = The Nine Point Circle Center of ABC] Which point is the NPC center N1 Tue, 14 Jul 2009 12:14:34 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/128 http://tech.groups.yahoo.com/group/Anopolis/message/128 1. Toledo Point ... [snip] ... 2. Let ABC be a triangle and A1,B1,C1 the NPC centers of OBC,OCA,OAB, resp. [where O = The Circumcenter of ABC] Which point is Mon, 13 Jul 2009 18:57:54 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/127 http://tech.groups.yahoo.com/group/Anopolis/message/127 ... [snip] ... Variation: Let ABC be a triangle and A1,B1,C1 the NPC centers of OBC,OCA,OAB, resp. [where O = The Circumcenter of ABC] Which point is the NPC Sat, 04 Jul 2009 11:28:53 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/126 http://tech.groups.yahoo.com/group/Anopolis/message/126 Let ABC be a triangle, P = (x:y:z) a point and A'B'C' the pedal triangle of P. Denote: A* := (The Reflection of BC in BP) /\ (The Reflection of BC in CP) B* := Mon, 29 Jun 2009 15:47:56 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/117 http://tech.groups.yahoo.com/group/Anopolis/message/117 Dear Antreas, [APH] ... Looks like the barycentrics for the DGAPH point is a^2 (49 a^8 - 133 a^6 b^2 + 120 a^4 b^4 - 29 a^2 b^6 - 7 b^8 - 133 a^6 c^2 + 21 a^4 Mon, 29 Jun 2009 06:13:46 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/116 http://tech.groups.yahoo.com/group/Anopolis/message/116 Dear Antreas, [APH] ... Looks like the barycentrics for the DGAPH point is a^2 (49 a^8 - 133 a^6 b^2 + 120 a^4 b^4 - 29 a^2 b^6 - 7 b^8 - 133 a^6 c^2 + 21 a^4 Sun, 28 Jun 2009 11:47:55 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/115 http://tech.groups.yahoo.com/group/Anopolis/message/115 Ac := The Orthogonal Projection of A' on AB aC := The Orthogonal Projection of A' on BB'. [The 4 points are collinear] 1. DON QUIXOTE POINT: La := The Euler Tue, 23 Jun 2009 07:37:46 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/112 http://tech.groups.yahoo.com/group/Anopolis/message/112 Let ABC be a triangle and A1,B1,C1 the NPC centers of NBC,NCA,NAB, resp. [ where N = The Nine Point Circle Center of ABC] Which point is the Circumcenter O1 of Sat, 20 Jun 2009 20:19:12 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/109 http://tech.groups.yahoo.com/group/Anopolis/message/109 Let ABC be a triangle, and A'B'C', A"B"C" the cevian, pedal triangles of I, resp. Denote: Ab := The Reflection of A in BB' (it is on BC) Ac := The Reflection Mon, 15 Jun 2009 18:53:32 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/108 http://tech.groups.yahoo.com/group/Anopolis/message/108 ... Let ABC be a triangle and A'B'C' the cevian triangle of I. Denote: Lab := The Reflection of AP in BP Lac := The Reflection of AP in CP Lbc := The Sun, 14 Jun 2009 17:49:00 GMT xpolakis http://tech.groups.yahoo.com/group/Anopolis/message/107 http://tech.groups.yahoo.com/group/Anopolis/message/107 (I) ... (II) Let ABC be a triangle and A'B'C' the cevian triangle of I. Denote: Lba := The Reflection of BI in AI Lca := The Reflection of CI in AI Lcb := The