[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the pedal triangle of I.
Denote:
Oa, Ob, Oc = the circumcenters of IBC, ICA, IAB, resp.
La, Lb, Lc = the Euler lines of IBC, ICA, IAB, resp.
Lab, Lac = the parallels through Oa to Lb, Lc, resp.
Lbc, Lba = the parallels through Ob to Lc, La, resp.
Lca, Lcb = the parallels through Oc to La, Lb, resp.
A* = Lbc /\ Lcb
B* = Lca /\ Lac
C* = Lab /\ Lba
1, A'B'C', A*B*C* are homothetic.
Homothetic center?
2. ABC, A*B*C* are orthologic.
Orthologic center (ABC, A*B*C*) = I
Orthologic center (A*B*C*, ABC) = ?
3. The orthocenter of A*B*C* lies on the Euler line of ABC.
Point?
[Ercole Suppa]:
Homotetic center(A'B'C',A*B*C*) =
MIDPOINT OF X(5441) AND X(16152) =
= a (b+c-1) (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c-3 a^3 b c-4 a^2 b^2 c+2 b^4 c-2 a^3 c^2-4 a^2 b c^2-2 a b^2 c^2-b^3 c^2+2 a^2 c^3-b^2 c^3+a c^4+2 b c^4-c^5) : : (barys)
= (24 a R^2-12 b R^2+4 b SB+c SB+b SC+4 c SC-4 a SW+3 b SW)S^2 + 4 R S^3+3 b SB SC^2-3 c SB SC^2-3 b SB SC SW : : (barys)
= 2*X[3647]-3*X[16370], 4*X[6701]-3*X[17532], X[6938]+X[16116]
= lies on these lines: {1,30}, {11,26725}, {21,60}, {35,16139}, {55,758}, {56,10122}, {65,3651}, {78,18253}, {191,3601}, {200,21677}, {354,18444}, {390,17482}, {442,1837}, {950,11263}, {958,31938}, {997,15670}, {1697,16126}, {1749,7508}, {2475,3486}, {2771,10058}, {3057,3957}, {3065,5424}, {3255,6596}, {3576,5427}, {3584,12738}, {3612,5428}, {3647,16370}, {3652,5693}, {3671,10123}, {3746,14988}, {3872,4863}, {3962,4640}, {4305,18977}, {4313,14450}, {4666,11281}, {5426,13384}, {5698,15677}, {5919,10698}, {6701,17532}, {6841,11375}, {6938,16116}, {7675,17768}, {7680,17718}, {10572,14526}, {10950,31419}, {11604,12743}, {12635,20835}, {12688,21669}, {12740,14100}, {15726,16133}, {17606,31254}, {17728,18443}
= midpoint of X(i) and X(j) for these {i,j}: {5441,16152}, {6938,16116}
= reflection of X(i) in X(j) for these {i,j}: {1836,3649}, {3652,6914}, {11684,4640}, {17637,10391}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,16132,3649},{21,17637,16141},{2646,17637,21}
= (6-9-13) search numbers: [2.3594864275788697229, 2.3577211193343425490, 0.9194022788703559279]
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Orthologic center (A*B*C*,ABC) =
MIDPOINT OF X(1) AND X(16132) =
= a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+a^4 b c-a^3 b^2 c+3 a b^4 c-b^5 c-a^4 c^2-a^3 b c^2+4 a^2 b^2 c^2-a b^3 c^2-b^4 c^2+4 a^3 c^3-a b^2 c^3+2 b^3 c^3-a^2 c^4+3 a b c^4-b^2 c^4-2 a c^5-b c^5+c^6) : : (barys)
=(20 a R^2-20 b R^2+4 b SB-c SB-b SC+4 c SC-4 a SW+5 b SW)S^2 + 4 R S SB SC+5 b SB SC^2-5 c SB SC^2-5 b SB SC SW : : (barys)
= 2*X[5]-3*X[26725], X[40]+X[16126], X[191]-3*X[3576], X[355]-2*X[442], X[944]+X[2475], X[1482]+X[16117], 5*X[3616]-2*X[22798], 2*X[3647]-X[13465], 3*X[3653]-2*X[15670], 3*X[5426]-X[7701], X[5690]-2*X[11277], 3*X[5731]+X[14450], 2*X[5901]-X[16160], X[6264]+X[13146], 4*X[6701]-X[18525], 2*X[8261]-3*X[10202], 4*X[9956]-5*X[31254], 2*X[12104]-X[19919], X[15680]+X[16116], 2*X[21677]-3*X[26446], 2*X[26202]-3*X[28461]
= lies on these: {1,30}, {3,758}, {5,26725}, {10,12738}, {21,104}, {35,11571}, {40,16126}, {100,13145}, {140,6326}, {191,3576}, {355,442}, {381,30143}, {515,11263}, {517,3651}, {548,5538}, {944,2475}, {952,5499}, {956,31938}, {960,13151}, {991,29097}, {997,18253}, {999,10122}, {1006,5694}, {1319,17637}, {1482,16117}, {1483,11014}, {2646,13369}, {3579,4018}, {3616,22798}, {3647,13465}, {3653,15670}, {3654,3811}, {3754,18524}, {3916,4511}, {3957,11278}, {4653,5492}, {5426,7701}, {5690,11277}, {5731,14450}, {5787,5886}, {5882,12737}, {5885,6905}, {5901,16160}, {6001,24299}, {6175,28204}, {6264,13146}, {6598,6907}, {6675,8583}, {6701,18525}, {6853,9803}, {6912,31828}, {6914,15071}, {6924,15016}, {6940,22935}, {7489,31803}, {7986,19765}, {8261,10202}, {9856,15178}, {9956,31254}, {10052,16142}, {10246,12114}, {10572,13273}, {10902,14988}, {11012,24475}, {12005,22765}, {12104,19919}, {12709,24929}, {12740,30538}, {15680,16116}, {15911,28186}, {16113,30264}, {16120,30283}, {16133,30284}, {16141,21842}, {16151,18454}, {21677,26446}, {26202,28461}, {26285,31660}, {27086,32612}, {28174,31651}
= midpoint of X(i) and X(j) for these {i,j}: {1,16132}, {40,16126}, {1482,16117}, {15680,16116}, {16159,18481}
= reflection of X(i) in X(j) for these {i,j}: {21,1385}, {191,5428}, {355,442}, {3652,21}, {5690,11277}, {6841,11281}, {7701,31649}, {11684,22937}, {13465,3647}, {16138,13743}, {16139,3}, {16159,3649}, {16160,5901}, {19919,12104}, {22937,13624}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {191,3576,5428}, {1385,21740,6265}, {1385,26201,104}, {3649,7354,79}, {5426,7701,31649}, {6841,11281,5886}, {11684,21161,22937}, {13465,28443,3647}, {13624,22937,21161}, {18444,21740,1385}
= (6-9-13) search numbers: [5.2955045433384276644, 5.2859939451710375150, -2.4629488078286904062]
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Orthocenter (A*B*C*) = X(361)
Best regards,
Ercole Suppa
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