[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point.
Denote:
1. Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
The reflections of AP, BP, CP in NbNc, NcNa, NaNb, resp.are concurrent at the Poncelet point of ABCP
2. Ha, Hb, Hc = the orthocenters of PBC, PCA, PAB, resp.
Which is the locus of P such that the reflections of AP, BP, CP in HbHc, HcHa, HaHb, resp. are concurrent?
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[Ercole Suppa]
*** The locus of P such that the reflections of AP, BP, CP in HbHc, HcHa, HaHb, resp. are concurrent is: {line AB} U {line BC} U {line CA} U {q7: circular circumseptic}
-- q7: ∑ c^2 (a^8-6 a^4 b^4+8 a^2 b^6-3 b^8-5 a^6 c^2+3 a^4 b^2 c^2+a^2 b^4 c^2+b^6 c^2+9 a^4 c^4-6 a^2 b^2 c^4-3 b^4 c^4-7 a^2 c^6+3 b^2 c^6+2 c^8) x^4 y^2 z + 4 (a^2-b^2) c^2 (a^2+b^2-c^2) (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) x^3 y^3 z + c^2 (3 a^8-8 a^6 b^2+6 a^4 b^4-b^8-a^6 c^2-a^4 b^2 c^2-3 a^2 b^4 c^2+5 b^6 c^2+3 a^4 c^4+6 a^2 b^2 c^4-9 b^4 c^4-3 a^2 c^6+7 b^2 c^6-2 c^8) x^2 y^4 z + (b^2-c^2)(a^8-a^6 b^2-3 a^4 b^4+5 a^2 b^6-2 b^8-a^6 c^2+3 a^2 b^4 c^2-2 b^6 c^2-3 a^4 c^4+3 a^2 b^2 c^4+8 b^4 c^4+5 a^2 c^6-2 b^2 c^6-2 c^8) x^3 y^2 z^2 + 2 a^6 (a^2-b^2-c^2) (a^2-b^2+c^2) y^4 z^3 + 2 a^6 (b^2+c^2-a^2) (a^2+b^2-c^2) y^3 z^4 = 0
-- q7 passes through ETC points X(i) for these i: 3,4,74,265
-- pairs {P=X(i) ∈ q7, Q=X(j)} for these {i,j}: {4,4}
-- Some points:
Q(X(3)) = X(2)X(49) ∩ X(3)X(68)
= (a^2-b^2-c^2) (a^8-2 a^6 b^2+2 a^4 b^4-2 a^2 b^6+b^8-2 a^6 c^2-2 a^4 b^2 c^2+2 a^2 b^4 c^2-4 b^6 c^2+2 a^4 c^4+2 a^2 b^2 c^4+6 b^4 c^4-2 a^2 c^6-4 b^2 c^6+c^8) : : (barys)
= (3 R^2+SB+SC-SW) S^2 + (R^2-SW) SB SC : : (barys)
= X[30]-X[11457], X[539]-X[1092], X[542]-X[10539], X[1204]-X[17702], X[1503]-X[7517], X[3564]-X[11585], X[13754]-X[18404]
= lies on these lines: {2,49}, {3,68}, {4,94}, {5,5422}, {6,5576}, {26,3580}, {30,11457}, {52,18381}, {54,23293}, {66,1351}, {69,3519}, {70,1993}, {125,1147}, {140,18911}, {155,2072}, {156,7505}, {182,1209}, {184,5449}, {185,9927}, {193,11255}, {195,15141}, {378,12370}, {381,11432}, {382,13093}, {389,18474}, {427,13292}, {539,1092}, {542,10539}, {569,21243}, {631,2888}, {858,16266}, {1181,10024}, {1204,17702}, {1352,1656}, {1498,11799}, {1503,7517}, {1568,15083}, {1594,12161}, {1614,10201}, {2070,9833}, {3090,3410}, {3357,16003}, {3521,15077}, {3542,10540}, {3548,6193}, {3549,6776}, {3564,11585}, {3567,11818}, {3767,22146}, {5094,15120}, {5446,11550}, {5654,10255}, {5889,18569}, {5946,7544}, {6143,9545}, {6243,6515}, {6288,9827}, {6643,23039}, {6644,14516}, {7383,13339}, {7391,10263}, {7394,10095}, {7401,18950}, {7506,12134}, {7526,12022}, {7528,11433}, {7529,18440}, {7530,16659}, {7689,21659}, {8541,12585}, {9140,18281}, {9544,14940}, {9707,10020}, {10055,18447}, {10071,18455}, {10112,13352}, {10264,11250}, {10298,12254}, {10605,12293}, {10620,20427}, {10627,16063}, {10984,18128}, {11264,13561}, {11411,18436}, {11412,14791}, {11456,15761}, {11572,14831}, {12121,18931}, {12162,18390}, {12163,18396}, {12421,23307}, {12902,18565}, {13579,14111}, {13598,14864}, {13754,18404}, {15027,15128}, {15089,16867}, {15749,17505}, {15760,18914}, {18394,18568}
= reflection of X11441) in X(5)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,18951,568}, {5,18356,11442}, {68,1899,3}, {125,1147,6640}, {184,5449,6639}, {1181,14852,10024}, {3548,6193,22115}, {5449,10116,184}, {6102,6746,568}, {6146,12359,3}, {6193,23291,3548}, {6515,14790,6243}, {10112,20299,13352}, {11411,18531,18436}, {11442,18912,5}, {12134,13567,7506}, {12163,18396,18563}
= (6-8-13) search numbers [3.62516865336754170, 4.22870331815668140, -0.960054116832188810]
Q(X(74)) = MIDPOINT OF X(3153) AND X(3448)
= a^10-2 a^8 b^2+a^6 b^4-a^4 b^6+2 a^2 b^8-b^10-2 a^8 c^2+a^6 b^2 c^2+a^4 b^4 c^2-3 a^2 b^6 c^2+3 b^8 c^2+a^6 c^4+a^4 b^2 c^4+2 a^2 b^4 c^4-2 b^6 c^4-a^4 c^6-3 a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8-c^10 : : (barys)
= (6 R^2-3 SW)SB SC + (4 R^2+SB+SC-SW)S^2 : : (barys)
= X[30]-X[74], X[110]-2*X[2072], 2*X[125]-X[186], X[323]-X[539], X[403]-X[1503], X[542]-X[1568], X[924]-X[5962], X[1154]-X[7574], 2*X[1495]-5*X[15081], 2*X[1531]+X[12317], X[2071]-X[17702], X[2777]-X[13399], X[3153]+X[3448], X[3564]-X[11416], 8*X[5159]-5*X[15034], X[5663]-X[18403], 2*X[7575]-5*X[15027], 4*X[10257]-3*X[15035], X[10296]+2*X[16003], 4*X[10297]-X[14094], X[10721]-X[15311], X[11563]-2*X[11801], X[12902]+X[18859], X[13619]-2*X[21663], X[14917]-X[18304], 7*X[15036]-8*X[16976], X[15054]+2*X[18323], 3*X[15061]-2*X[15646]
= lies on these lines: {2,11464}, {3,12278}, {4,51}, {5,1614}, {20,9927}, {22,14852}, {30,74}, {49,10224}, {50,1157}, {54,1594},{67,11564}, {68,70}, {110,2072}, {115,13509}, {125,186}, {140,6288}, {156,10255}, {184,7577}, {195,11264}, {232,15340}, {235,16659}, {323,539}, {378,1853}, {381,5422}, {403,1503}, {427,12022}, {542,1568}, {858,15133}, {924,5962}, {1141,3484}, {1154,7574}, {1181,7547}, {1199,3574}, {1216,2888}, {1352,12283}, {1478,19368}, {1479,11461}, {1495,15081}, {1531,12317}, {1596,16658}, {1650,6760}, {1656,9707}, {2071,17702}, {2777,13399}, {2914,10114}, {2979,14791}, {3153,3448}, {3410,5891}, {3520,20299}, {3541,15126}, {3545,3618}, {3564,11416}, {3843,12174}, {5159,15034}, {5449,7488}, {5576,13434}, {5640,11818}, {5663,18403}, {5889,18569}, {5899,13171}, {6030,25337}, {6143,12254}, {6240,15138}, {6247,18560}, {6640,11449}, {6643,7999}, {6759,16868}, {6776,7699}, {7401,11465}, {7505,9833}, {7507,7592}, {7533,14845}, {7544,15024}, {7575,15027}, {7576,13567}, {7579,18432}, {7687,10821}, {7722,19506}, {7741,9638}, {9705,9820}, {10113,17854}, {10149,12904}, {10257,15035}, {10282,14940}, {10296,16003}, {10297,14094}, {10304,18387}, {10605,18405}, {10610,11565}, {10721,15311}, {11413,12293}, {11438,18559}, {11440,18563}, {11442,11459}, {11466,18582}, {11467,18581}, {11563,11801}, {11585,14516}, {12006,22804}, {12111,18404}, {12225,12359}, {12241,15559}, {12902,18859}, {13403,14865}, {13491,18379}, {13619,21663}, {14483,15321}, {14561,19123}, {14917,18304}, {15032,18388}, {15036,16976}, {15043,18952}, {15045,18420}, {15054,18323}, {15061,15646}, {15072,18392}, {16072,18440}, {16661,17712}, {18356,18436}, {18533,23291}, {18914,23047}
= midpoint of X(i) and X(j) for these {i,j}: {3153,3448}, {10733,13445}
= reflection of X(i) in X(j) for these {i,j}: {4,13851}, {110,2072}, {186,125}, {10540,5}, {11563,11801}, {13619,21663}, {14157,403}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,1899,5890}, {4,11457,6241}, {4,14216,12290}, {4,18912,3567}, {184,23325,7577}, {185,18383,4}, {389,11572,4}, {427,12022,15033}, {1594,6146,54}, {1853,18396,378}, {5449,11750,7488}, {6143,12254,13367}, {6241,18394,4}, {11442,18531,11459}, {11550,18390,4}, {12289,23294,3}, {14157,14644,403}, {18381,18390,11550}, {18420,18911,15045}, {20299,21659,3520}
= (6-8-13) search numbers [3.47682364652544924, 3.53451984456388496, -0.410998631956210135]
Q(X(265)) = X(30)X(12219) ∩ X(2072)X(3564)
= (b^2+c^2-a^2) (2 a^14-7 a^12 b^2+8 a^10 b^4-a^8 b^6-6 a^6 b^8+7 a^4 b^10-4 a^2 b^12+b^14-7 a^12 c^2+12 a^10 b^2 c^2-9 a^8 b^4 c^2+10 a^6 b^6 c^2-11 a^4 b^8 c^2+10 a^2 b^10 c^2-5 b^12 c^2+8 a^10 c^4-9 a^8 b^2 c^4-4 a^6 b^4 c^4+4 a^4 b^6 c^4-4 a^2 b^8 c^4+9 b^10 c^4-a^8 c^6+10 a^6 b^2 c^6+4 a^4 b^4 c^6-4 a^2 b^6 c^6-5 b^8 c^6-6 a^6 c^8-11 a^4 b^2 c^8-4 a^2 b^4 c^8-5 b^6 c^8+7 a^4 c^10+10 a^2 b^2 c^10+9 b^4 c^10-4 a^2 c^12-5 b^2 c^12+c^14) : : (barys)
= (3 R^4+3 R^2 SW-SW^2)SB SC + (5 R^4-8 R^2 SB-8 R^2 SC+3 R^2 SW+2 SB SW+2 SC SW-SW^2) S^2 : : (barys)
= lies on these lines: {30,12219}, {2072,3564}, {3167,6640}, {9820,24572}, {12429,18404}
= (6-8-13) search numbers [1.34917902860406625, -2.14582617980149596, 4.50353843933738419]
Best regards
Ercole Suppa
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