A", B", C" = the isogonal conjugates of Ia, Ib, Ic wrt triangle A'B'C', resp.
1. ABC, A"B"C" are perspective
2. A'B'C', A"B"C" are perspective.
[César Lozada]:
1) X(3059)
2) Z2 = X(9)X(55) ∩ X(11)X(142)
= a*(-a+b+c)*((b+c)*a^2+(b-c)^2* (-2*a+b+c)) : : (barycentrics)
= 2*X(7)-3*X(354) = X(7)-3*X(7671) = 4*X(9)-3*X(210) = 3*X(210)-2*X(3059) = 4*X(2550)-5*X(3698) = 2*X(2951)-3*X(5918) = X(3962)-4*X(5698)
= On lines: {1,971}, {4,12710}, {6,4319}, {7,354}, {9,55}, {10,9844}, {11,142}, {33,608}, {37,2293}, {40,10398}, {44,1253}, {56,5732}, {57,2951}, {65,516}, {72,4314}, {144,145}, {181,10443}, {224,1001}, {226,7965}, {241,1742}, {527,3058}, {528,12743}, {938,9943}, {942,4312}, {946,12671}, {954,1898}, {960,4313}, {1058,12675}, {1100,4336}, {1155,1445}, {1156,2346}, {1317,2801}, {1386,3100}, {1418,3000}, {1479,5805}, {1486,2182}, {1697,5223}, {1699,11018}, {1708,7964}, {1721,5228}, {1827,1839}, {1837,2550}, {1852,1890}, {2098,3243}, {3085,5817}, {3254,3255}, {3295,5779}, {3304,4321}, {3488,6001}, {3555,5850}, {3601,5696}, {3616,10861}, {3649,4890}, {3666,4335}, {3740,5281}, {3742,5274}, {3748,8545}, {3753,12690}, {3893,5853}, {4003,7004}, {5173,9580}, {5432,6666}, {5435,10178}, {5693,9957}, {5735,9670}, {5759,7957}, {5880,10940}, {5927,13405}, {6172,10385}, {6173,11238}, {7354,12573}, {8257,11502}, {9355,9440}, {10167,11019}, {10202,10738}, {10442,10473}, {10572,13375}, {10578,10865}, {11496,12664}
= midpoint of X(390) and X(10394)
= reflection of X(i) in X(j) for these (i,j): (7,5572), (65,5728), (354,7671), (3057,390), (3059,9), (4312,942), (5784,1001), (7354,12573), (7957,5759), (8581,1), (12688,11372)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1, 9848, 10866), (1, 12680, 9850), (7, 5572, 354), (7, 7671, 5572), (9, 3059, 210), (9, 3174, 480), (9, 4326, 55), (55, 1864, 210), (480, 3174, 3689), (497, 10391, 354), (2293, 2310, 37), (8581, 10384, 10866)
= [ 2.422021374406637, 2.52505904842289, 0.774690660196241 ]
César Lozada
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