[Antreas P. Hatzipolakis]:
Let ABC be a triangle, A'B'C' the pedal triangle of O, P a point and PaPbPc the pedal triangle of P.
Denote:
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
The line A'N intersects again (Na) at A*
The line B'N intersects again (Nb) at B*
The line C'N intersects again (Nc) at C*
Which is the locus of P such that PaPbPc, A*B*C* are
1. perspective ?
N lies on the locus (Hyacinthos 28345)
2. orthologic?
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[Ercole Suppa]
(1)
*** Γ₂ = locus of points P such that PaPbPc, A*B*C* are perspective = the entire plane
-- Q1(X(i)) = isogonal conjugate of X(i) with respect to the pedal triangle of X(i)
-- ETC pairs {P, Q1(P)} : {1,65},{2,5640},{3,5},{4,4},{5,10095},{6,18907},{8,14923},{15,5472},{16,5471},{20,12111},{36,20118},{40,14872},{98,13137},{99,12833},{110,7471},{946,10110}
-- some points Q1(X(i)) :
P1=Q1(X(7)) = X(7)X(517)∩X(57)X(934) =
isogonal conjugate of X(7) with respect to the pedal triangle of X(7)
= a (a+b-c)^2 (a-b+c)^2 (a b-b^2+a c-3 b c-c^2) :: (barys)
= lies on these lines : {7,517},{57,934},{65,279},{77,11529},{85,3869},{269,18421},{664,3873},{942,3160},{994,3668},{1088,4566},{1159,1443},{1170,2082},{1323,5902},{1442,15934},{1446,3212},{1462,1572},{2809,7672},{3339,7177},{3340,4350},{3868,9312},{4328,9819},{5543,9957},{5903,10481},{6516,9352},{6604,14923},{6767,7269}
= ETC - (6,9,13) search numbers: [0.281415496818196286, 0.411670669631189996, 3.22577763516977236]
P2=Q1(X(9)) = X(9)X(374)∩X(40)X(15288) = isogonal conjugate of X(9) with respect to the pedal triangle of X(9)
= a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-6 a^4 b c+6 a^3 b^2 c-4 a^2 b^3 c+a b^4 c+2 b^5 c-a^4 c^2+6 a^3 b c^2-4 a^2 b^2 c^2-2 a b^3 c^2+b^4 c^2-2 a^3 c^3-4 a^2 b c^3-2 a b^2 c^3-4 b^3 c^3+2 a^2 c^4+a b c^4+b^2 c^4+a c^5+2 b c^5-c^6) :: (barys)
= lies on these lines : {9,374},{40,15288},{65,169},{72,22011},{101,354},{198,18443},{226,4904},{1212,14110},{1903,4846},{2389,21867},{2809,5728},{3057,16601},{3730,7957},{3753,8074},{5819,6001},{6554,7686}
= ETC - (6,9,13) search numbers: [1.31030415503588826, 0.364964097621961001, 2.78324126584491921]
P3=Q1(X(10)) = MIDPOINT OF X(355) AND X(389)
= -a^2 (a^3 b^2+a^2 b^3-a b^4-b^5+a^2 b^2 c-b^4 c+a^3 c^2+a^2 b c^2+4 b^3 c^2+a^2 c^3+4 b^2 c^3-a c^4-b c^4-c^5) :: (barys)
= X[1]-3*X[5943], 3*X[2]+X[16980], X[8]+3*X[51], X[40]+X[13598], X[52]+3*X[5790] ,X[145]-9*X[5640], X[355]+X[389], 9*X[373]-5*X[3616], 3*X[375]-X[960], 2*X[1125]-3*X[6688], X[1385]-2*X[11695], X[1483]-5*X[15026], 5*X[1698]-3*X[3819], 3*X[3060]+5*X[3617] ,7*X[3622]-15*X[11451], X[3679]+X[21849], X[3751]+X[14913], 3*X[3917]-7*X[9780], X[4297]-2*X[17704], 7*X[4678]+9*X[11002], X[5446]+X[5690] ,X[5562]-5*X[5818],3*X[5587]-X[5907],9*X[5650]-13*X[19877], 2*X[6684]-X[13348], 3*X[7967]-11*X[15024], 3*X[9730]+X[18525],7*X[9781]+X[12245],2*X[9956]-X[11793], 6*X[10219]-5*X[19862], 9*X[12045]-8*X[19878], X[12237]+X[12787], X[12238]+X[12788], 3*X[13570]-2*X[18483], 9*X[14845]-5*X[18493],3*X[16836]-X[18481]
= lies on these lines: {1,5943},{2,16980},{8,51},{10,511},{40,13598},{52,5790},{101,1126},{145,5640},{181,5247},{182,9798},{197,13323},{355,389},{373,3616},{375,960},{515,9729},{517,5795},{518,9822},{674,4662},{916,9947},{942,2810},{952,5462},{958,970},{993,15489},{1125,6688},{1385,11695},{1469,1722},{1483,15026},{1698,3819},{2390,10107},{2392,3918},{2551,10441},{2807,19925},{3060,3617},{3271,5255},{3295,4266},{3622,11451},{3679,21849},{3686,9052},{3751,14913},{3812,8679},{3917,9780},{4245,5399},{4297,17704},{4663,8681},{4678,11002},{5260,22076},{5293,10544},{5302,22276},{5446,5690},{5562,5818},{5587,5907},{5650,19877},{5752,9708},{5844,10095},{6000,18480},{6684,13348},{7967,15024},{8192,10601},{9730,18525},{9781,12245},{9956,11793},{10219,19862},{10459,20962},{12045,19878},{12237,12787},{12238,12788},{12410,17810},{13570,18483},{13754,18357},{14845,18493},{16836,18481},{17757,18180}
= midpoint of X(i) and X(j) for these {i,j}: {355,389},{3679,21849},{3751,14913},{5446,5690},{12237,12787},{12238,12788}
= reflection of X(i) in X(j) for these {i,j}: {1385,11695},{4297,17704},{11793,9956},{13348,6684}
= ETC - (6,9,13) search numbers: [1.21734152666864423, 0.352409999707091287, 2.83483839287855703]
---------------------------
(2)
*** Γ₂ = locus of points P such that PaPbPc, A*B*C* are orthologic =
= {Linf} ∪ {excentral-circum-circular quintic Q038 } ∪ {q7=circum septic}
-- Γ₂ pass through vertices A,B,C, through Ia, Ib, Ic excenters, through Ha, Hb, Hc feet of altitudes
-- ETC centers X(i) on Q038 : 1, 4, 5, 80, 1113, 1114, 1263, 2009, 2010
-- ETC pairs {P ∈ Q038 , Q2(P)} : {4,4},{5,6153}
-- other properties of Q038 are on: Q038
--- equation of q7 = a^6 c^4 x^3 y^2-a^4 b^2 c^4 x^3 y^2-a^2 b^4 c^4 x^3 y^2+b^6 c^4 x^3 y^2-2 a^4 c^6 x^3 y^2-2 b^4 c^6 x^3 y^2+a^2 c^8 x^3 y^2+b^2 c^8 x^3 y^2+a^6 c^4 x^2 y^3-a^4 b^2 c^4 x^2 y^3-a^2 b^4 c^4 x^2 y^3+b^6 c^4 x^2 y^3-2 a^4 c^6 x^2 y^3-2 b^4 c^6 x^2 y^3+a^2 c^8 x^2 y^3+b^2 c^8 x^2 y^3+a^10 x^3 y z-4 a^8 b^2 x^3 y z+6 a^6 b^4 x^3 y z-4 a^4 b^6 x^3 y z+a^2 b^8 x^3 y z-4 a^8 c^2 x^3 y z+9 a^6 b^2 c^2 x^3 y z -5 a^4 b^4 c^2 x^3 y z-a^2 b^6 c^2 x^3 y z+b^8 c^2 x^3 y z+6 a^6 c^4 x^3 y z-5 a^4 b^2 c^4 x^3 y z-b^6 c^4 x^3 y z-4 a^4 c^6 x^3 y z-a^2 b^2 c^6 x^3 y z-b^4 c^6 x^3 y z+a^2 c^8 x^3 y z+b^2 c^8 x^3 y z+a^10 x^2 y^2 z-3 a^8 b^2 x^2 y^2 z+2 a^6 b^4 x^2 y^2 z+2 a^4 b^6 x^2 y^2 z-3 a^2 b^8 x^2 y^2 z+b^10 x^2 y^2 z-2 a^8 c^2 x^2 y^2 z+4 a^6 b^2 c^2 x^2 y^2 z-4 a^4 b^4 c^2 x^2 y^2 z+4 a^2 b^6 c^2 x^2 y^2 z-2 b^8 c^2 x^2 y^2 z +2 a^6 c^4 x^2 y^2 z-2 a^4 b^2 c^4 x^2 y^2 z-2 a^2 b^4 c^4 x^2 y^2 z+2 b^6 c^4 x^2 y^2 z-a^4 c^6 x^2 y^2 z+2 a^2 b^2 c^6 x^2 y^2 z-b^4 c^6 x^2 y^2 z-a^2 c^8 x^2 y^2 z-b^2 c^8 x^2 y^2 z+c^10 x^2 y^2 z+a^8 b^2 x y^3 z-4 a^6 b^4 x y^3 z+6 a^4 b^6 x y^3 z-4 a^2 b^8 x y^3 z+b^10 x y^3 z+a^8 c^2 x y^3 z-a^6 b^2 c^2 x y^3 z-5 a^4 b^4 c^2 x y^3 z+9 a^2 b^6 c^2 x y^3 z-4 b^8 c^2 x y^3 z-a^6 c^4 x y^3 z-5 a^2 b^4 c^4 x y^3 z+6 b^6 c^4 x y^3 z-a^4 c^6 x y^3 z-a^2 b^2 c^6 x y^3 z-4 b^4 c^6 x y^3 z+a^2 c^8 x y^3 z+b^2 c^8 x y^3 z+a^6 b^4 x^3 z^2-2 a^4 b^6 x^3 z^2+a^2 b^8 x^3 z^2-a^4 b^4 c^2 x^3 z^2+b^8 c^2 x^3 z^2-a^2 b^4 c^4 x^3 z^2-2 b^6 c^4 x^3 z^2+b^4 c^6 x^3 z^2+a^10 x^2 y z^2-2 a^8 b^2 x^2 y z^2+2 a^6 b^4 x^2 y z^2-a^4 b^6 x^2 y z^2 -a^2 b^8 x^2 y z^2+b^10 x^2 y z^2-3 a^8 c^2 x^2 y z^2+4 a^6 b^2 c^2 x^2 y z^2-2 a^4 b^4 c^2 x^2 y z^2+2 a^2 b^6 c^2 x^2 y z^2-b^8 c^2 x^2 y z^2+2 a^6 c^4 x^2 y z^2-4 a^4 b^2 c^4 x^2 y z^2-2 a^2 b^4 c^4 x^2 y z^2-b^6 c^4 x^2 y z^2+2 a^4 c^6 x^2 y z^2+4 a^2 b^2 c^6 x^2 y z^2+2 b^4 c^6 x^2 y z^2-3 a^2 c^8 x^2 y z^2-2 b^2 c^8 x^2 y z^2+c^10 x^2 y z^2+a^10 x y^2 z^2-a^8 b^2 x y^2 z^2-a^6 b^4 x y^2 z^2+2 a^4 b^6 x y^2 z^2-2 a^2 b^8 x y^2 z^2+b^10 x y^2 z^2-a^8 c^2 x y^2 z^2+2 a^6 b^2 c^2 x y^2 z^2-2 a^4 b^4 c^2 x y^2 z^2+4 a^2 b^6 c^2 x y^2 z^2-3 b^8 c^2 x y^2 z^2-a^6 c^4 x y^2 z^2-2 a^4 b^2 c^4 x y^2 z^2-4 a^2 b^4 c^4 x y^2 z^2+2 b^6 c^4 x y^2 z^2+2 a^4 c^6 x y^2 z^2+4 a^2 b^2 c^6 x y^2 z^2+2 b^4 c^6 x y^2 z^2-2 a^2 c^8 x y^2 z^2-3 b^2 c^8 x y^2 z^2+c^10 x y^2 z^2+a^8 b^2 y^3 z^2-2 a^6 b^4 y^3 z^2+a^4 b^6 y^3 z^2+a^8 c^2 y^3 z^2-a^4 b^4 c^2 y^3 z^2-2 a^6 c^4 y^3 z^2-a^4 b^2 c^4 y^3 z^2+a^4 c^6 y^3 z^2+a^6 b^4 x^2 z^3-2 a^4 b^6 x^2 z^3+a^2 b^8 x^2 z^3-a^4 b^4 c^2 x^2 z^3+b^8 c^2 x^2 z^3-a^2 b^4 c^4 x^2 z^3-2 b^6 c^4 x^2 z^3+b^4 c^6 x^2 z^3+a^8 b^2 x y z^3-a^6 b^4 x y z^3-a^4 b^6 x y z^3+a^2 b^8 x y z^3+a^8 c^2 x y z^3-a^6 b^2 c^2 x y z^3-a^2 b^6 c^2 x y z^3 +b^8 c^2 x y z^3-4 a^6 c^4 x y z^3-5 a^4 b^2 c^4 x y z^3-5 a^2 b^4 c^4 x y z^3-4 b^6 c^4 x y z^3+6 a^4 c^6 x y z^3+9 a^2 b^2 c^6 x y z^3+6 b^4 c^6 x y z^3-4 a^2 c^8 x y z^3-4 b^2 c^8 x y z^3+c^10 x y z^3+a^8 b^2 y^2 z^3-2 a^6 b^4 y^2 z^3+a^4 b^6 y^2 z^3+a^8 c^2 y^2 z^3-a^4 b^4 c^2 y^2 z^3-2 a^6 c^4 y^2 z^3 -a^4 b^2 c^4 y^2 z^3+a^4 c^6 y^2 z^3
-- ETC centers X(i) on q7 : 265
-- ETC pairs {P ∈ q7 , Q2(P)} : {265,523}
*** some points Q2(X(i))
P4=Q1(X(1)) = X(1)X(7335) ∩ X(56)X(11334)
= -a (a+b-c) (a-b+c) (2 a^3-a^2 b-2 a b^2+b^3-a^2 c+2 a b c-b^2 c-2 a c^2-b c^2+c^3) (a^4-a^3 b+a b^3-b^4-a^3 c+2 a^2 b c-a b^2 c-a b c^2+2 b^2 c^2+a c^3-c^4) : : (barys)
= lies on these lines: {1,7335},{56,11334},{65,11700},{946,1319}
= (6,9,13) search numbers: [0.0852284451650059128, 0.728819846958723399, 3.09676069009064479]
Best regards
Ercole Suppa
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