[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A1B1C1, A2B2C2 the pedal, antipedal triangles of P, resp.
Denote:
D = the Poncelet point of ABCP
For P = I (D = X(11)):
1. The parallels to DA1, DB1, DC1 through A2, B2, C2, resp. are concurrent.
2. The parallels to DA2, DB2, DC2 through A1, B1, C1, resp. are concurrent.
Points?
Loci (if not complicated :) ?
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[Ercole Suppa]
(1)
*** Locus of point P such that the parallels to DA1, DB1, DC1 through A2, B2, C2, are councurrent: {Linf} U {circumcircle} U {circum-quartic q4} U {q8 excentral-circum-curve of degree 8}
-- q4 pass through vertices A,B,C, and through X(4)=orthocenter; equation
q4: c^4 x^2 y^2+a^4 x^2 y z-2 a^2 b^2 x^2 y z+b^4 x^2 y z-2 a^2 c^2 x^2 y z+c^4 x^2 y z+a^4 x y^2 z-2 a^2 b^2 x y^2 z+b^4 x y^2 z-2 b^2 c^2 x y^2 z +c^4 x y^2 z+b^4 x^2 z^2+a^4 x y z^2+b^4 x y z^2-2 a^2 c^2 x y z^2-2 b^2 c^2 x y z^2+c^4 x y z^2+a^4 y^2 z^2 = 0
-- q8 pass through vertices A,B,C, through excenters Ia,Ib,Ic and through ETC points X(1) and X(4)
q8: -2 b^2 c^6 x^5 y^3-2 a^2 c^6 x^4 y^4-2 b^2 c^6 x^4 y^4+2 c^8 x^4 y^4-2 a^2 c^6 x^3 y^5-a^4 b^2 c^2 x^5 y^2 z+2 a^2 b^4 c^2 x^5 y^2 z-b^6 c^2 x^5 y^2 z +4 a^2 b^2 c^4 x^5 y^2 z-2 b^4 c^4 x^5 y^2 z-3 b^2 c^6 x^5 y^2 z-a^6 c^2 x^4 y^3 z+3 a^2 b^4 c^2 x^4 y^3 z-2 b^6 c^2 x^4 y^3 z+4 a^4 c^4 x^4 y^3 z+2 b^4 c^4 x^4 y^3 z -5 a^2 c^6 x^4 y^3 z-2 b^2 c^6 x^4 y^3 z+2 c^8 x^4 y^3 z-2 a^6 c^2 x^3 y^4 z+3 a^4 b^2 c^2 x^3 y^4 z-b^6 c^2 x^3 y^4 z+2 a^4 c^4 x^3 y^4 z+4 b^4 c^4 x^3 y^4 z -2 a^2 c^6 x^3 y^4 z-5 b^2 c^6 x^3 y^4 z+2 c^8 x^3 y^4 z-a^6 c^2 x^2 y^5 z+2 a^4 b^2 c^2 x^2 y^5 z-a^2 b^4 c^2 x^2 y^5 z-2 a^4 c^4 x^2 y^5 z+4 a^2 b^2 c^4 x^2 y^5 z -3 a^2 c^6 x^2 y^5 z-a^4 b^2 c^2 x^5 y z^2+4 a^2 b^4 c^2 x^5 y z^2-3 b^6 c^2 x^5 y z^2+2 a^2 b^2 c^4 x^5 y z^2-2 b^4 c^4 x^5 y z^2-b^2 c^6 x^5 y z^2+2 a^4 b^2 c^2 x^4 y^2 z^2 -2 b^6 c^2 x^4 y^2 z^2+4 b^4 c^4 x^4 y^2 z^2-2 b^2 c^6 x^4 y^2 z^2+a^6 c^2 x^3 y^3 z^2-a^4 b^2 c^2 x^3 y^3 z^2-a^2 b^4 c^2 x^3 y^3 z^2+b^6 c^2 x^3 y^3 z^2+8 a^2 b^2 c^4 x^3 y^3 z^2 -3 a^2 c^6 x^3 y^3 z^2-3 b^2 c^6 x^3 y^3 z^2+2 c^8 x^3 y^3 z^2-2 a^6 c^2 x^2 y^4 z^2+2 a^2 b^4 c^2 x^2 y^4 z^2+4 a^4 c^4 x^2 y^4 z^2-2 a^2 c^6 x^2 y^4 z^2-3 a^6 c^2 x y^5 z^2 +4 a^4 b^2 c^2 x y^5 z^2-a^2 b^4 c^2 x y^5 z^2-2 a^4 c^4 x y^5 z^2+2 a^2 b^2 c^4 x y^5 z^2-a^2 c^6 x y^5 z^2-2 b^6 c^2 x^5 z^3-a^6 b^2 x^4 y z^3+4 a^4 b^4 x^4 y z^3 -5 a^2 b^6 x^4 y z^3+2 b^8 x^4 y z^3-2 b^6 c^2 x^4 y z^3+3 a^2 b^2 c^4 x^4 y z^3+2 b^4 c^4 x^4 y z^3-2 b^2 c^6 x^4 y z^3+a^6 b^2 x^3 y^2 z^3-3 a^2 b^6 x^3 y^2 z^3 +2 b^8 x^3 y^2 z^3-a^4 b^2 c^2 x^3 y^2 z^3+8 a^2 b^4 c^2 x^3 y^2 z^3-3 b^6 c^2 x^3 y^2 z^3-a^2 b^2 c^4 x^3 y^2 z^3+b^2 c^6 x^3 y^2 z^3+2 a^8 x^2 y^3 z^3-3 a^6 b^2 x^2 y^3 z^3 +a^2 b^6 x^2 y^3 z^3-3 a^6 c^2 x^2 y^3 z^3+8 a^4 b^2 c^2 x^2 y^3 z^3-a^2 b^4 c^2 x^2 y^3 z^3-a^2 b^2 c^4 x^2 y^3 z^3+a^2 c^6 x^2 y^3 z^3+2 a^8 x y^4 z^3-5 a^6 b^2 x y^4 z^3 +4 a^4 b^4 x y^4 z^3-a^2 b^6 x y^4 z^3-2 a^6 c^2 x y^4 z^3+2 a^4 c^4 x y^4 z^3+3 a^2 b^2 c^4 x y^4 z^3-2 a^2 c^6 x y^4 z^3-2 a^6 c^2 y^5 z^3-2 a^2 b^6 x^4 z^4+2 b^8 x^4 z^4 -2 b^6 c^2 x^4 z^4-2 a^6 b^2 x^3 y z^4+2 a^4 b^4 x^3 y z^4-2 a^2 b^6 x^3 y z^4+2 b^8 x^3 y z^4+3 a^4 b^2 c^2 x^3 y z^4-5 b^6 c^2 x^3 y z^4+4 b^4 c^4 x^3 y z^4-b^2 c^6 x^3 y z^4 -2 a^6 b^2 x^2 y^2 z^4+4 a^4 b^4 x^2 y^2 z^4-2 a^2 b^6 x^2 y^2 z^4+2 a^2 b^2 c^4 x^2 y^2 z^4+2 a^8 x y^3 z^4-2 a^6 b^2 x y^3 z^4+2 a^4 b^4 x y^3 z^4-2 a^2 b^6 x y^3 z^4 -5 a^6 c^2 x y^3 z^4+3 a^2 b^4 c^2 x y^3 z^4+4 a^4 c^4 x y^3 z^4-a^2 c^6 x y^3 z^4+2 a^8 y^4 z^4-2 a^6 b^2 y^4 z^4-2 a^6 c^2 y^4 z^4-2 a^2 b^6 x^3 z^5-a^6 b^2 x^2 y z^5 -2 a^4 b^4 x^2 y z^5-3 a^2 b^6 x^2 y z^5+2 a^4 b^2 c^2 x^2 y z^5+4 a^2 b^4 c^2 x^2 y z^5-a^2 b^2 c^4 x^2 y z^5-3 a^6 b^2 x y^2 z^5-2 a^4 b^4 x y^2 z^5-a^2 b^6 x y^2 z^5 +4 a^4 b^2 c^2 x y^2 z^5+2 a^2 b^4 c^2 x y^2 z^5-a^2 b^2 c^4 x y^2 z^5-2 a^6 b^2 y^3 z^5 = 0
*** Let Q=Q(P) denote the point of concurrence.
-- If P ∈ Linf then Q(P) ∈ Linf
-- If P ∈ circumcircle then Q(P) ∈ cubic q3 of equation
q3: (-18 b^2 c^4+2 c^6) x^3+(9 a^2 c^4-27 b^2 c^4+3 c^6) x^2 y+(27 a^2 c^4-9 b^2 c^4-3 c^6) x y^2+(18 a^2 c^4-2 c^6) y^3+(9 a^2 b^2 c^2-9 b^4 c^2+6 a^2 c^4-15 b^2 c^4) x^2 z +(9 a^4 c^2-9 b^4 c^2+15 a^2 c^4-15 b^2 c^4) x y z+(9 a^4 c^2-9 a^2 b^2 c^2+15 a^2 c^4-6 b^2 c^4) y^2 z+(6 a^4 c^2+6 a^2 b^2 c^2-12 b^4 c^2) x z^2+(12 a^4 c^2-6 a^2 b^2 c^2 -6 b^4 c^2) y z^2+(2 a^6-6 a^4 b^2+6 a^2 b^4-2 b^6) z^3 = 0
-- If P=X(1)=I then Q1=Q(X(1)) = X(1768)
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(2)
*** Locus of point P such that the parallels to DA2, DB2, DC2 through A1, B1, C1, are councurrent:
{Linf} U {circumcircle} U {circum-quartic q4 given above} U {degree 12 circum-excentral curve through ETC points X(1) and X(4)}
q12: equation is complicated
-- If P ∈ circumcircle Q(P) ∈ cubic q3 of equation
q3: (-18 b^2 c^4+2 c^6) x^3+(9 a^2 c^4-27 b^2 c^4+3 c^6) x^2 y+(27 a^2 c^4-9 b^2 c^4-3 c^6) x y^2+(18 a^2 c^4-2 c^6) y^3+(9 a^2 b^2 c^2-9 b^4 c^2+6 a^2 c^4-15 b^2 c^4) x^2 z +(9 a^4 c^2-9 b^4 c^2+15 a^2 c^4-15 b^2 c^4) x y z+(9 a^4 c^2-9 a^2 b^2 c^2+15 a^2 c^4-6 b^2 c^4) y^2 z+(6 a^4 c^2+6 a^2 b^2 c^2-12 b^4 c^2) x z^2 +(12 a^4 c^2-6 a^2 b^2 c^2-6 b^4 c^2) y z^2+(2 a^6-6 a^4 b^2+6 a^2 b^4-2 b^6) z^3 = 0
-- If P ∈ Linf then Q(P) ∈ Linf
-- If P=I then
Q3=Q(X(1)) = MIDPOINT OF X(4292) AND X(12736)
Barycentrics 2 a^6-2 a^5 (b+c)+2 a^3 (b-c)^2 (b+c)-(b-c)^4 (b+c)^2+a^4 (-3 b^2+8 b c-3 c^2)+2 a^2 (b-c)^2 (b^2-b c+c^2) : :
= lies on these lines: {5,7702},{7,100},{11,57},{30,18838},{56,1387},{65,952},{80,3339},{109,1086},{149,21454},{214,3671},{226,3035},{388,1145},{516,3660},{528,553},{938,10724},{942,5840},{1155,5762},{1317,3340},{1320,3600},{1466,10090},{1470,11729},{1537,4295},{1617,3474},{2099,11046},{2802,4298},{2829,4292},{2834,3937},{3036,4848},{3218,5857},{3333,14217},{3336,8068},{3337,5533},{3361,16173},{3911,5087},{4355,5541},{4440,14594},{4654,6174},{4860,13274},{5221,12019},{5708,10738},{5854,10106},{6147,10044},{7972,18421},{9945,12739},{10404,10956},{10742,18541},{11529,12119},{14151,20095},{15803,21154},{17579,18419}
= midpoint of X(4292) and X(12736)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {109,1086,15253},{5221,13273,12832},{12832,13273,12019}
= (6,9,13) search numbers: [0.638310542768662302, 0.923716737529187990, 2.70656341310940482]
Best regards
Ercole Suppa
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