[Antreas P. Hatzipolakis]:
Let ABC be a triangle, AhBhCh the pedal triangle of H and O' the Poncelet point of ABCO [ O' = X125]
(ie O' is the point the NPCs of ABC, OBC, OCA, OAB are concurrent at).
Denote:
hA = the second intersection of AhO' and the NPC of OBC [other than O']
hB = the second intersection of BhO' and the NPC of OCA
hC = the second intersection of ChO' and the NPC of OAB
O', hA, hB, hC are concyclic
Which point is the center of the circle?
GENERALIZATION (Conjecture):
Let ABC be a triangle, D, D* two isogonal conjugate points, AdBdCd the pedal triangle of D and D' the Poncelet point of ABCD*
(ie D' is the point the NPCs of ABC, D*BC, D*CA, D*AB are concurrent at).
Denote:
dA = the second intersection of AdD' and the NPC of D*BC [other than D']
dB = the second intersection of BdD' and the NPC of D*CA
dC = the second intersection of CdD' and the NPC of D*AB
D', dA, dB, dC are concyclic.
Call this circle (ABC, D). Which is its center in terms (of the coordinates) of D?
Now, in a quadrangle:
Let ABCD be a quadrangle and A*, B*, C*, D* the isogonal conjugates of A,B,C,D wrt triangles BCD, CDA, DAB, ABC, resp.
We have two tetrads of circles:
(ABC, D), (BCD, A), (CDA, B), (DAB, C)
(ABC, D*), (BCD, A*), (CDA, B*), (DAB, C*)
Which properties have these circles?
If no one can be found, then at least Cayley–Bacharach theorem can be applied:
The cubics passing through the eight centers of the circles pass through a fixed point.
[César Lozada]:
> GENERALIZATION (Conjecture):
> D', dA, dB, dC are concyclic.
Let D=u:v:w (trilinears). Then the center Z(D) of the circle {D’, dA, dB, dC} is:
Z(D) = (v^2+w^2)*a*b*c*(b*c*u^2 - SA*w*v) + 2*a*(S^2-SA^2)*v^2*w^2 + b*c*u*(SB*b*w^3 + SC*c*v^3) + u*v*w*((3*S^2+SA*SC)*c*v + (3*S^2+SA*SB)*b*w + 2*a*b*c*SA*u) : :
If D is on the circumcircle, Z(D)=isogonal(D)
If D is in the infinity, Z(D) = Midpoint(0, isogonal(P)), so Z(D) is on the circle (O, R/2). ETC pair (D, Z(D)): (523, 1511)
Other ETC pairs (D, Z(D)) : (1,1), (3,5), (15,5459), (16,5460), (36,11)
Some non-ETC :
Z(G) = midpoint of X(6) and X(8542)
= (2*a^6-3*(b^2+c^2)*a^4-2*(b^4+ b^2*c^2+c^4)*a^2+(b^2-3*c^2)*( 3*b^2-c^2)*(b^2+c^2))*a : : (trilinears)
= X(6)+X(8542) = 3*X(182)-X(8547) = X(8547)+9*X(9813)
= on lines: {6,373}, {39,9145}, {182,2393}, {193,7605}, {523,7804}, {524,547}, {575,2854}, {576,10170}, {597,5972}, {1843,2916}, {3618,5486}, {5092,8705}, {5650,10510}, {9730,11579}, {11003,11188}
= midpoint of X(6) and X(8542)
= [ 2.377903015171476, 0.84411325894432, 1.958784680251474 ]
Z(H) = midpoint of X(3) and X(1147)
= (2*a^6-3*(b^2+c^2)*a^4+2*b^2* c^2*a^2+(b^4-c^4)*(b^2-c^2))*( -a^2+b^2+c^2) *a : : (trilinears)
= cos(A)*(1+2*cos(2*A)+cos(2*B)+ cos(2*C)) : : (trilinears)
= 3*X(2)-X(9927) = 3*X(3)+X(155) = X(3)+X(1147) = 5*X(3)+3*X(3167) = 3*X(3)-X(7689) = X(20)+3*X(5654) = X(155)-3*X(1147) = 5*X(155)-9*X(3167) = X(155)+X(7689) = 5*X(1147)-3*X(3167)
= On lines: {2,9927}, {3,49}, {4,11449}, {5,1511}, {20,5654}, {24,5446}, {26,11202}, {30,5448}, {52,186}, {54,5504}, {68,631}, {74,9705}, {110,3520}, {140,5449}, {156,6000}, {182,8548}, {378,10539}, {382,1495}, {511,1658}, {539,549}, {541,5894}, {550,5944}, {567,2931}, {569,5892}, {575,12006}, {578,5462}, {858,11750}, {1069,5217}, {1152,8909}, {1614,2071}, {3043,11562}, {3157,5204}, {3517,12002}, {3523,6193}, {3524,11411}, {3530,3564}, {3576,9928}, {3855,10546}, {5010,6238}, {5646,7393}, {5657,9933}, {5663,10226}, {5890,9545}, {6146,10257}, {6200,10666}, {6241,9544}, {6396,10665}, {6418,8912}, {6642,11425}, {6689,7399}, {6699,10116}, {7280,7352}, {7488,10625}, {7503,10170}, {7506,11424}, {7514,9938}, {7526,9306}, {7575,10263}, {8546,8681}, {9707,11413}, {10020,10182}, {10298,11412}, {10540,11381}, {10645,10662}, {10646,10661}
= midpoint of X(i) and X(j) for these {i,j}: {3,1147}, {155,7689}, {156,11250}
= reflection of X(i) in X(j) for these (i,j): (5448,9820), (5449,140)
= complement of X(9927)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,49,185), (3,155,7689), (3,1092,1216), (578,6644,5462), (1147,7689,155), (1614,2071,10575)
= [ 5.020692065572603, 3.50252376074738, -1.101402151951162 ]
Z(K) = midpoint of X(2) and X(11165)
= (8*a^4-17*(b^2+c^2)*a^2-2*b^2* c^2+5*c^4+5*b^4)/a : : (trilinears)
= 5*X(2)-X(5485) = 7*X(2)+X(11148) = X(2)+X(11165) = X(5)-2*X(9771) = 2*X(140)-X(7610) = 3*X(549)-2*X(5569) = X(549)-2*X(7622) = 7*X(5485)+5*X(11148) = X(5485)+5*X(11165) = X(5569)-3*X(7622) = X(7618)+X(11184) = X(11148)-7*X(11165)
= on lines: {2,2418}, {3,9770}, {5,543}, {30,7618}, {39,9167}, {83,5503}, {99,3363}, {140,7610}, {182,524}, {538,7619}, {547,7615}, {550,7775}, {597,620}, {631,9740}, {1007,5077}, {2482,3815}, {2549,8355}, {3845,8176}, {3849,8703}, {5013,8360}, {5055,7620}, {5215,5306}, {7763,8359}, {7769,9166}, {7777,8598}, {7870,8362}, {8182,9766}, {8667,11812}
= midpoint of X(i) and X(j) for these {i,j}: {2,11165}, {3,9770}, {7615,8716}, {7618,11184}, {8182,9766}
= reflection of X(i) in X(j) for these (i,j): (5,9771), (549,7622), (3845,8176), (7610,140), (7615,547)
= [ -0.273892582550908, 3.02194816081169, 1.674958485599784 ]
Z( X(30) ) = midpoint of X(3) and X(74)
= (2*a^8-3*(b^2+c^2)*a^6-3*(b^4- 4*b^2*c^2+c^4)*a^4+(b^2+c^2)*( 7*b^4-15*b^2*c^2+7*c^4)*a^2-( 3*b^4+7*b^2*c^2+3*c^4)*(b^2-c^ 2)^2)*a : : (trilinears)
= (3*cos(2*A)+7/2)*cos(B-C)-6* cos(A)-cos(3*A) : : (trilinears)
= 3*X(2)-X(7728) = X(3)+X(74) = 3*X(3)-X(110) = 5*X(3)-X(399) = 2*X(3)-X(1511) = 4*X(3)-X(5609) = 3*X(3)+X(10620) = 2*X(5)-X(1539) = X(5)-2*X(6699) = 3*X(5)-4*X(6723) = 3*X(74)+X(110) = 5*X(74)+X(399) = X(1539)-4*X(6699) = 3*X(1539)-8*X(6723) = 3*X(6699)-2*X(6723)
= on lines: {2,7728}, {3,74}, {5,1539}, {20,265}, {30,125}, {35,3028}, {55,10081}, {56,10065}, {64,9934}, {113,140}, {146,631}, {182,2781}, {185,10226}, {376,3448}, {378,1112}, {381,10721}, {511,11806}, {517,11709}, {541,549}, {542,8703}, {550,10264}, {567,1986}, {974,1204}, {1154,2071}, {1350,5621}, {1351,5622}, {1657,10733}, {2420,3269}, {2771,9943}, {2780,9208}, {2854,3098}, {2935,7526}, {3521,6143}, {3524,5655}, {3530,10272}, {3532,5504}, {3534,9140}, {3576,9904}, {3581,7464}, {3627,7687}, {3818,6698}, {5050,10752}, {5054,10706}, {5085,9970}, {5092,6593}, {5204,10091}, {5217,10088}, {5462,11807}, {5544,9818}, {6101,7689}, {6409,10819}, {6410,10820}, {6642,9919}, {6644,10117}, {6689,11805}, {7280,7727}, {7502,8717}, {7583,8994}, {7722,11003}, {7731,10574}, {7978,10246}, {8718,11559}, {9729,11557}, {10610,10628}, {11438,11746}
= midpoint of X(i) and X(j) for these {i,j}: {3,74}, {20,265}, {64,9934}, {110,10620}, {113,10990}, {550,10264}, {1350,11579}, {1657,10733}, {3534,9140}, {3581,7464}, {8718,11559}
= reflection of X(i) in X(j) for these (i,j): (5,6699), (113,140), (1511,3), (1539,5), (3627,7687), (3818,6698), (5609,1511), (6102,974), (6593,5092), (10113,125), (10272,3530), (11557,9729), (11702,10610), (11805,6689), (11807,5462)
= complement of X(7728)
= circumcircle-inverse-of-X( 10620)
= [ 9.719762854243672, 9.29279598626787, -7.278854056698148 ]
Z( X(511) ) = midpoint of X(3) and X(98)
= (2*a^8-3*(b^2+c^2)*a^6+3*(b^4+ c^4)*a^4-(b^2+c^2)*(2*b^4-3*b^ 2*c^2+2*c^4)*a^2-(b^2-c^2)^2* b^2*c^2)/a : : )trilinears)
= 3*X(2)-X(6033) = X(3)+X(98) = 3*X(3)-X(99) = X(5)-2*X(6036) = 3*X(5)-4*X(6722) = X(20)+X(6321) = 3*X(98)+X(99) = X(114)-2*X(140) = X(114)+X(10991) = X(115)-3*X(6055) = 2*X(140)+X(10991) = 3*X(6036)-2*X(6722)
= on lines: {2,5191}, {3,76}, {5,2794}, {20,6321}, {30,115}, {32,2023}, {35,3027}, {36,3023}, {55,10069}, {56,10053}, {114,140}, {141,542}, {147,631}, {148,376}, {157,1605}, {182,10007}, {262,11842}, {378,5186}, {381,3972}, {404,5985}, {517,11710}, {543,8703}, {550,11623}, {632,6721}, {671,3534}, {1657,10723}, {1916,7793}, {2080,5999}, {2784,6684}, {3095,7766}, {3098,5969}, {3111,5663}, {3329,3398}, {3523,5984}, {3524,8289}, {3576,9860}, {3830,9166}, {3845,5461}, {4027,7824}, {5027,11176}, {5050,10753}, {5054,6054}, {5149,7815}, {5182,12017}, {5204,10089}, {5217,10086}, {5569,9830}, {5961,7502}, {5986,7485}, {5987,7496}, {6642,9861}, {6671,6771}, {6672,6774}, {7583,8980}, {7776,8781}, {7798,9737}, {7857,9873}, {7970,10246}, {8667,9888}, {8725,11606}, {9167,11812}, {10352,11285}
= midpoint of X(i) and X(j) for these {i,j}: {3,98}, {20,6321}, {114,10991}, {376,11632}, {671,3534}, {1657,10723}, {1916,9821}, {2080,5999}, {6033,9862}, {6295,6582}, {8667,9888}, {8724,11177}, {8725,11606}
= reflection of X(i) in X(j) for these (i,j): (5,6036), (114,140), (3845,5461), (5026,5092)
= complement of X(6033)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2,9862,6033), (1078,5152,5976), (3524,11177,8724)
= [ 8.454179614470462, 9.60084291120245, -6.908001970988301 ]
César Lozada
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