[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
Nab, Nac = the reflections of Nb, Nc in PA'
M1 = the midpoint of NabNac
Similarly M2, M3
ABC, M1M2M3 are orthologic.
APH
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[Ercole Suppa]:
Let P(u:v:w) be the barycentric coordinates of P. Denote: Z1(P) = orthology center(ABC,M1M2M3), Z2(P) = orthology center(M1M2M3,ABC). We have
Z1(P) = a^2 (-a^2 c^2 u^2 v^2+b^2 c^2 u^2 v^2+c^4 u^2 v^2+2 a^4 u^2 v w-2 a^2 b^2 u^2 v w-4 a^2 c^2 u^2 v w-b^2 c^2 u^2 v w+2 c^4 u^2 v w-a^4 u v^2 w+4 a^2 b^2 u v^2 w-3 b^4 u v^2 w+2 a^2 c^2 u v^2 w+4 b^2 c^2 u v^2 w-c^4 u v^2 w-b^4 u^2 w^2+2 a^4 u v w^2-a^2 b^2 u v w^2-4 a^2 c^2 u v w^2-2 b^2 c^2 u v w^2+2 c^4 u v w^2+a^4 v^2 w^2+a^2 b^2 v^2 w^2-a^2 c^2 v^2 w^2) (-c^4 u^2 v^2+2 a^4 u^2 v w-4 a^2 b^2 u^2 v w+2 b^4 u^2 v w-2 a^2 c^2 u^2 v w-b^2 c^2 u^2 v w+2 a^4 u v^2 w-4 a^2 b^2 u v^2 w+2 b^4 u v^2 w-a^2 c^2 u v^2 w-2 b^2 c^2 u v^2 w-a^2 b^2 u^2 w^2+b^4 u^2 w^2+b^2 c^2 u^2 w^2-a^4 u v w^2+2 a^2 b^2 u v w^2-b^4 u v w^2+4 a^2 c^2 u v w^2+4 b^2 c^2 u v w^2-3 c^4 u v w^2+a^4 v^2 w^2-a^2 b^2 v^2 w^2+a^2 c^2 v^2 w^2) : : (barys)
Z2(P) = a^2 c^2 u^2 v^2-b^2 c^2 u^2 v^2+c^4 u^2 v^2-8 a^4 u^2 v w+13 a^2 b^2 u^2 v w-5 b^4 u^2 v w+13 a^2 c^2 u^2 v w+10 b^2 c^2 u^2 v w-5 c^4 u^2 v w-a^4 u v^2 w-a^2 b^2 u v^2 w+2 b^4 u v^2 w-a^2 c^2 u v^2 w-4 b^2 c^2 u v^2 w+2 c^4 u v^2 w+a^2 b^2 u^2 w^2+b^4 u^2 w^2-b^2 c^2 u^2 w^2-a^4 u v w^2-a^2 b^2 u v w^2+2 b^4 u v w^2-a^2 c^2 u v w^2-4 b^2 c^2 u v w^2+2 c^4 u v w^2-2 a^4 v^2 w^2 : : (barys)
Some points:
*** P = X(1) = I
Z1(P) = ISOGONAL CONJUGATE OF X(9897)
= a^2 (2 a^4-4 a^2 b^2+2 b^4-3 a^3 c+3 a^2 b c+3 a b^2 c-3 b^3 c+a^2 c^2-5 a b c^2+b^2 c^2+3 a c^3+3 b c^3-3 c^4) (2 a^4-3 a^3 b+a^2 b^2+3 a b^3-3 b^4+3 a^2 b c-5 a b^2 c+3 b^3 c-4 a^2 c^2+3 a b c^2+b^2 c^2-3 b c^3+2 c^4) : : (barys)
= lies on these lines: {484,4511}, {2323,19297}
= barycentric quotient X(6)/X(9897)
= trilinear quotient X(1)/X(9897)
= (6-9-13) search numbers: [-14.57549664052077037785, -9.52352022698980617354, 16.96102308852536970535]
Z2(P) = X(1)X(381) ∩ X(3)X(3633)
= -8 a^4+7 a^3 b+6 a^2 b^2-7 a b^3+2 b^4+7 a^3 c-14 a^2 b c+7 a b^2 c+6 a^2 c^2+7 a b c^2-4 b^2 c^2-7 a c^3+2 c^4 : : (barys)
= 7*X[1]-3*X[381], 3*X[3]+X[3633], 7*X[8]-15*X[631], 7*X[10]-9*X[11539], 3*X[140]-2*X[4691], X[145]+X[3579], 3*X[355]-7*X[3622], 3*X[549]-X[3625], 7*X[944]+X[3146], X[1071]+X[10284], 5*X[1482]-X[9589], X[1657]+3*X[11224], 5*X[1698]-9*X[10246], 7*X[3241]+X[11001], 9*X[3524]-X[20053], 17*X[3544]-21*X[5886], 5*X[3617]-9*X[3653], 7*X[3632]-27*X[15707], 2*X[3636]-X[18357], 7*X[3654]-11*X[15715], 3*X[3656]-7*X[20057], 5*X[4668]-9*X[5054], 9*X[5657]-X[20054], X[5881]-3*X[11230], 3*X[5885]-2*X[10107], 2*X[6684]-3*X[31662], 3*X[10247]-X[22793], 3*X[11231]-X[12645], 7*X[12702]-15*X[15695], 9*X[17502]+X[20014]
= lies on these lines: {1,381}, {3,3633}, {8,631}, {10,11539}, {30,3635}, {79,11011}, {140,4691}, {145,3579}, {355,3622}, {515,3853}, {517,550}, {519,12100}, {549,3625}, {572,4727}, {944,3146}, {950,12735}, {952,1125}, {971,26200}, {1071,10284}, {1317,2771}, {1482,9589}, {1657,11224}, {1698,10246}, {2099,31776}, {2646,7972}, {3241,11001}, {3524,20053}, {3544,5886}, {3617,3653}, {3632,15707}, {3636,18357}, {3654,15715}, {3656,20057}, {3746,12773}, {3847,10942}, {3880,13145}, {4301,28168}, {4668,5054}, {5657,20054}, {5844,31663}, {5881,11230}, {5885,10107}, {5901,28236}, {6684,31662}, {10106,16137}, {10247,22793}, {10950,25405}, {11231,12645}, {12702,15695}, {13464,28224}, {17502,20014}, {22791,28208}, {22935,24927}
= midpoint of X(i) and X(j) for these {i,j}: {145,3579}, {944,10222}, {1071,10284}, {1483,5882}, {11278,18481}
= reflection of X(i) in X(j) for these {i,j}: {9955,1}, {9956,15178}, {13369,26089}, {15178,13607}, {18357,3636}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,18526,18480}, {145,3655,3579}, {944,3623,12699 {3241,18481,11278}, {3623,12699,10222}
= (6-9-13) search numbers: [4.10075887346398981046, 4.74988429410964212624, -1.54037489407490689680]
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*** P = X(2)
Z1(P) = X(8593)X(15533) ∩ X(9208)X(11643)
= a^2 (2 a^4-4 a^2 b^2+2 b^4+a^2 c^2+b^2 c^2-2 c^4) (2 a^4+a^2 b^2-2 b^4-4 a^2 c^2+b^2 c^2+2 c^4) : : (barys)
= lies on these lines: {8593,15533}, {9208,11643}
= (6-9-13) search numbers: [-9.63001201936541105156, -8.13895974792304780056, 13.71994985479206345038]
Z2(P) = X(3)
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*** P = X(3)
Z1(P) = (name pending)
= a^2 (2 a^10-6 a^8 b^2+4 a^6 b^4+4 a^4 b^6-6 a^2 b^8+2 b^10-2 a^8 c^2+7 a^6 b^2 c^2-10 a^4 b^4 c^2+7 a^2 b^6 c^2-2 b^8 c^2-3 a^6 c^4+3 a^4 b^2 c^4+3 a^2 b^4 c^4-3 b^6 c^4+a^4 c^6-9 a^2 b^2 c^6+b^4 c^6+5 a^2 c^8+5 b^2 c^8-3 c^10) (2 a^10-2 a^8 b^2-3 a^6 b^4+a^4 b^6+5 a^2 b^8-3 b^10-6 a^8 c^2+7 a^6 b^2 c^2+3 a^4 b^4 c^2-9 a^2 b^6 c^2+5 b^8 c^2+4 a^6 c^4-10 a^4 b^2 c^4+3 a^2 b^4 c^4+b^6 c^4+4 a^4 c^6+7 a^2 b^2 c^6-3 b^4 c^6-6 a^2 c^8-2 b^2 c^8+2 c^10) : : (barys)
= lies on this line: {1511,5562}
= (6-9-13) search numbers: [6.03922023285546071821, 6.12721965661432199766, -3.38858923398959491905]
Z2(P) = MIDPOINT OF X(20) AND X(10282)
= 8 a^10-15 a^8 b^2-a^6 b^4+13 a^4 b^6-3 a^2 b^8-2 b^10-15 a^8 c^2+28 a^6 b^2 c^2-13 a^4 b^4 c^2-6 a^2 b^6 c^2+6 b^8 c^2-a^6 c^4-13 a^4 b^2 c^4+18 a^2 b^4 c^4-4 b^6 c^4+13 a^4 c^6-6 a^2 b^2 c^6-4 b^4 c^6-3 a^2 c^8+6 b^2 c^8-2 c^10 : : (barys)
= 3*X[3]-X[18383], X[20]+X[10282], X[64]-9*X[15689], 9*X[376]-X[14216], 3*X[548]-X[6696], X[1657]+3*X[11202], X[2883]+3*X[15686], X[3357]-5*X[15696], 5*X[3522]-X[18381], 7*X[3523]-3*X[18376], 7*X[3528]-3*X[23329], 9*X[3534]-X[5925], X[3627]-3*X[10182], X[6225]+15*X[17538], 3*X[8703]-X[20299], 3*X[11204]-X[14864], 3*X[15681]+5*X[17821]
= lies on these lines: {3,18383}, {20,10282}, {64,15689}, {376,14216}, {389,10295}, {548,6696}, {550,1216}, {1657,11202}, 2071,23358}, {2777,12103}, {2883,15686}, {3357,15696}, {3522,18381}, {3523,18376}, {3528,23329}, {3534,5925}, {3627,10182}, {6225,17538}, {8703,20299}, {10628,15644}, {11204,14864}, {13367,13619}, {13754,15332}, {13851,17506}, {15681,17821}
= midpoint of X(i) and X(j) for these {i,j}: {20,10282}, {14864,17845}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {11204,17845,14864}
= (6-9-13) search numbers: [9.90623390027736029093, 8.69287924747776430371, -6.94959064339747016449]
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*** P = X(4)
Z1(P) = X(74)
Z2(P) = X(3627)
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*** P = X(5)
Z1(P) = X(3)
Z2(P) = MIDPOINT OF X(20414) AND X(30484)
= 2 a^16-10 a^14 b^2+25 a^12 b^4-44 a^10 b^6+55 a^8 b^8-42 a^6 b^10+15 a^4 b^12-b^16-10 a^14 c^2+38 a^12 b^2 c^2-58 a^10 b^4 c^2+27 a^8 b^6 c^2+35 a^6 b^8 c^2-47 a^4 b^10 c^2+13 a^2 b^12 c^2+2 b^14 c^2+25 a^12 c^4-58 a^10 b^2 c^4+34 a^8 b^4 c^4+7 a^6 b^6 c^4+23 a^4 b^8 c^4-39 a^2 b^10 c^4+8 b^12 c^4-44 a^10 c^6+27 a^8 b^2 c^6+7 a^6 b^4 c^6+18 a^4 b^6 c^6+26 a^2 b^8 c^6-34 b^10 c^6+55 a^8 c^8+35 a^6 b^2 c^8+23 a^4 b^4 c^8+26 a^2 b^6 c^8+50 b^8 c^8-42 a^6 c^10-47 a^4 b^2 c^10-39 a^2 b^4 c^10-34 b^6 c^10+15 a^4 c^12+13 a^2 b^2 c^12+8 b^4 c^12+2 b^2 c^14-c^16 : : (barys)
= 3*X[5066]-X[32536], 7*X[14869]+X[15619], X[27868]+X[32551]
= lies on these lines: {5,6150}, {140,13470}, {252,3574}, {539,13856}, {3850,10275}, {5066,32536}, {10289,32423}, {14869,15619}, {27868,32551}
= midpoint of X(i) and X(j) for these {i,j}: {20414,30484}, {27868,32551}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {5,30484,20414}
= (6-9-13) search numbers: [3.64194256035922794333, 3.65668177479530267083, -0.57178100542432025823]
Best regards,
Ercole Suppa
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