[Antreas P; Hatzipolakis]:
Let ABC be a triangle and P a point.
Denote:
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
A', B', C' = the midpoints of AP, BP, CP, resp.
D = the Poncelet point of ABCP
Ab, Ac = the antipodes of A' in (Nb), (Nc), resp.
Bc, Ba = the antipodes of B' in (Nc), (Na), resp.
Ca, Cb = the antipodes of C' in (Na), (Nb), resp.
Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.
1. the perpendicular bisectors of BaCa, CbAb, AcBc concur at the midpoint of OP
2. Ma, Mb, Mc and D are concyclic.
Center of the circle in terms of P?
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[Ercole Suppa]
*** If P=(u:v:w) (barys) then the center of circle (Ma,Mb,Mc,D) is the point
Q(P)= -a^4 v^2 w^2 (2 u+v+w)+a^2 u (c^2 v (u^2 (v-w)-w^2 (v+w)+u (v^2-2 v w-2 w^2))+b^2 w (u^2 (-v+w)-v^2 (v+w)+u (-2 v^2-2 v w+w^2)))+u (b^4 w (u^2 v+v (v+w)^2+u (2 v^2+3 v w+w^2))+c^4 v (u^2 w+w (v+w)^2+u (v^2+3 v w+2 w^2))-b^2 c^2 (2 v w (v+w)^2+u^2 (v^2+4 v w+w^2)+u (v^3+5 v^2 w+5 v w^2+w^3))) : : (barys)
*** Pairs {P=X(i), Q(P)=X(j)} : {1,5},{2,20112},{3,13561},{4,5},{5,5501},{6,20113},{13,5459},{14,5460},{74,125},{80,12619},{98,115},{99,114},{100,119},{101,118},{102,124},{103,116},{104,11},{105,5511},{106,5510},{107,133},{108,25640},{109,117},{110,113},{111,5512},{112,132},{476,25641},{477,3258},{675,25642},{691,16188},{842,5099},{915,5521},{917,5190},{925,131},{930,128},{933,18402},{953,3259},{972,5514},{1113,1312},{1114,1313},{1138,20124},{1141,137},{1292,120},{1293,121},{1294,122},{1295,123},{1296,126},{1297,127},{1298,130},{1299,135},{1300,136},{1303,129},{1304,18809},{1379,2040},{1380,2039},{2373,14672},{2687,5520},{2693,16177},{2698,2679},{2723,15612},{2724,1566},{2734,10017},{3563,5139},{5606,5950},{5951,5952},{6082,6092},{6233,13234},{6323,12494},{10121,15169},{11568,13994},{12507,13249},{13238,12624},{13597,11792},{14720,21662},{15323,5518},{15324,13613},{18401,20625},{22751,14103},{23232,138},{23233,139}
*** Some points:
Q(X(7)) = X(11)X(7671) ∩ X(1001)X(1006)
= 2 a^7 b-7 a^6 b^2+6 a^5 b^3+5 a^4 b^4-10 a^3 b^5+3 a^2 b^6+2 a b^7-b^8+2 a^7 c-4 a^6 b c-6 a^4 b^3 c+24 a^3 b^4 c-18 a^2 b^5 c-2 a b^6 c+4 b^7 c-7 a^6 c^2+22 a^4 b^2 c^2-14 a^3 b^3 c^2+9 a^2 b^4 c^2-6 a b^5 c^2-4 b^6 c^2+6 a^5 c^3-6 a^4 b c^3-14 a^3 b^2 c^3+12 a^2 b^3 c^3+6 a b^4 c^3-4 b^5 c^3+5 a^4 c^4+24 a^3 b c^4+9 a^2 b^2 c^4+6 a b^3 c^4+10 b^4 c^4-10 a^3 c^5-18 a^2 b c^5-6 a b^2 c^5-4 b^3 c^5+3 a^2 c^6-2 a b c^6-4 b^2 c^6+2 a c^7+4 b c^7-c^8 : : (barys)
= lies on these lines: {11,7671}, {1001,1006}, {5851,5886}, {5856,5901}
= (6-8-13) search numbers [1.15147691419069392, 0.649411041731204183, 2.65962133877476363]
Q(X(8)) = X(11)X(8256) ∩ X(355)X(528)
= 2 a^6 b-3 a^5 b^2-3 a^4 b^3+6 a^3 b^4-3 a b^6+b^7+2 a^6 c-8 a^5 b c+15 a^4 b^2 c+2 a^3 b^3 c-16 a^2 b^4 c+6 a b^5 c-b^6 c-3 a^5 c^2+15 a^4 b c^2-36 a^3 b^2 c^2+20 a^2 b^3 c^2+3 a b^4 c^2-3 b^5 c^2-3 a^4 c^3+2 a^3 b c^3+20 a^2 b^2 c^3-12 a b^3 c^3+3 b^4 c^3+6 a^3 c^4-16 a^2 b c^4+3 a b^2 c^4+3 b^3 c^4+6 a b c^5-3 b^2 c^5-3 a c^6-b c^6+c^7 : : (barys)
= lies on these lines: {11,8256}, {355,528}, {1145,10826}, {2802,10943}, {3057,15842}, {3434,10953}, {3816,17622}, {3829,17619}, {3871,6224}, {3880,12616}, {3885,10949}, {5687,10043}, {8668,12114}, {10785,10912}, {10948,17652}, {12607,12672}
= (6-8-13) search numbers [-0.751571028397640211, -21.9775614446366446, 19.2027782643778901}]
Q(X(10)) = X(115)X(8258) ∩ X(3828)X(17677)
= 2 a^7+3 a^6 b+2 a^4 b^3+2 a^3 b^4-6 a^2 b^5-6 a b^6-b^7+3 a^6 c+4 a^5 b c+4 a^4 b^2 c+4 a^3 b^3 c-4 a^2 b^4 c-10 a b^5 c-5 b^6 c+4 a^4 b c^2+6 a^3 b^2 c^2+13 a^2 b^3 c^2+8 a b^4 c^2-2 b^5 c^2+2 a^4 c^3+4 a^3 b c^3+13 a^2 b^2 c^3+24 a b^3 c^3+12 b^4 c^3+2 a^3 c^4-4 a^2 b c^4+8 a b^2 c^4+12 b^3 c^4-6 a^2 c^5-10 a b c^5-2 b^2 c^5-6 a c^6-5 b c^6-c^7 : : (barys)
= lies on these lines: {115,8258}, {3828,17677}
= (6-8-13) search numbers [1.12267577358849295, -3.39753208563904215, 5.47464403030902692]
Best regards
Ercole Suppa
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