HYACINTHOS 24739

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

A"B"C" = the orthic triangle of A'B'C'

A*, B*, C* = the reflections of A", B", C" in IA', IB', IC', resp.

ABC, A*B*C* are orthologic.

Tangential triangle version:

Let ABC be a triangle and A'B'C' the pedal triangle of H.

Denote:

A"B"C" = the antipedal triangle of O.

A*, B*, C* = the reflections of A', B', C' in OA, OB, OC, resp.

A"B"C", A*B*C* are orthologic.

[César Lozada]:

 

1^st construction:

Orthologic centers (ABC, A*B*C*) =

= a/(a^4+(b+c)*a^3-2*a^2*b*c-(b^ 2-c^2)*(b-c)*a-(b^2-c^2)^2) : : (trilinears)

= On lines: {3,3681}, {35,603}, {48,3730}, {56,378}, {1437,4184}, {1442,3295}

= Trilinear pole of the line {6586,9404}

= [ 7.510548775075639, 7.53354192680340, -5.041271671299040 ]

 

Orthologic centers (A*B*C*, ABC) =

= (2*a^4+(b+c)*a^3-(b^2+6*b*c+c^ 2)*a^2-(b^2-c^2)*(b-c)*a-(b^2- c^2)^2)/a : : (trilinears)

= (6*R+r)*X(1)-(4*R+r)*X(7) = 3*X(1)-X(1770) = 3*X(1)-2*X(4298)

= On lines: {1,7}, {2,9614}, {4,1697}, {8,3586}, {9,5082}, {10,1479}, {11,6684}, {30,9957}, {35,404}, {40,497}, {46,5493}, {55,946}, {57,1058}, {65,3058}, {72,5853}, {80,3626}, {100,6700}, {140,7743}, {144,6764}, {149,6734}, {165,3086}, {226,3295}, {329,6765}, {355,9668}, {376,1420}, {389,517}, {452,9623}, {496,3579}, {498,3817}, {499,10164}, {515,3057}, {519,3869}, {527,3555}, {528,960}, {548,5126}, {551,3612}, {553,5045}, {908,3871}, {938,2093}, {944,7962}, {1056,9579}, {1361,2816}, {1367,3021}, {1490,10388}, {1496,1777}, {1497,1754}, {1698,10591}, {1699,3085}, {1706,5084}, {1737,4857}, {1836,3303}, {1837,9670}, {2078,3651}, {2098,5882}, {2136,3421}, {2792,10544}, {3146,9613}, {3159,4463}, {3333,3474}, {3340,3488}, {3419,5837}, {3436,3895}, {3452,5687}, {3485,10385}, {3486,7982}, {3487,10389}, {3583,10039}, {3601,5603}, {3634,7741}, {3635,5441}, {3649,3748}, {3710,5014}, {3717,5100}, {3813,4640}, {3878,6737}, {3914,3915}, {3947,10056}, {4652,10529}, {4848,5722}, {5046,6735}, {5173,10122}, {5217,10165}, {5218,8227}, {5223,9804}, {5225,5587}, {5267,10058}, {5316,9709}, {5657,9581}, {5691,9819}, {5692,6743}, {5759,10384}, {5902,6744}, {5919,7354}, {6745,8715}, {7173,10172}, {7264,10521}, {7672,10399}, {7682,10531}, {7957,9848}, {9612,9812}

= midpoint of X(i) and X(j) for these {i,j}: {3057,6284}, {5697,10572}

= reflection of X(i) in X(j) for these (i,j): (1770,4298), (4292,1), (6737,3878), (10106,9957)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1,20,4311), (1,1770,4298), (1,4294,4304), (1,4299,4315), (1,4302,4297), (1,4309,4314), (1,4333,4317), (1,9589,4295), (20,9785,1), (40,497,1210), (390,962,1), (496,3579,3911), (1479,5119,10), (1697,9580,4), (1770,4298,4292), (3434,5250,10), (4297,4342,1), (4301,4314,1)

= [ 3.690426650751870, 3.58555154221956, -0.544914270745408 ]

 

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Tangential triangle version

Orthologic centers (A*B*C*, A”B”C”) =Z* and  (A”B”C”, A*B*C*) =Z”, where

 

Z* = Reflection of X(52) in X(3)

= cos(2*A)*cos(B-C)+2*cos(A) : : (trilinears)

= 2*X(3)-X(52), 3*X(3)-2*X(389), 5*X(3)-3*X(568)

= On lines: {2,5446}, {3,6}, {4,1216}, {5,3917}, {20,6193}, {22,1147}, {26,1092}, {30,5562}, {51,140}, {54,6636}, {68,1370}, {141,7403}, {143,549}, {156,3292}, {185,550}, {323,1614}, {373,632}, {376,5889}, {382,5907}, {394,7387}, {427,1209}, {517,1770}, {548,6102}, {631,3060}, {858,5449}, {1595,1843}, {1656,3819}, {1657,5925}, {1993,10323}, {2888,5189}, {2937,10282}, {3072,7186}, {3073,3792}, {3088,6403}, {3090,7998}, {3091,7999}, {3520,6242}, {3522,5890}, {3523,3567}, {3525,5640}, {3526,5943}, {3528,10574}, {3530,5946}, {3628,5650}, {5480,7405}, {5944,7555}, {6030,9706}, {6689,7495}, {6923,10441}, {7404,10519}, {7492,9545}, {7517,9306}

= reflection of X(i) in X(j) for these (i,j): (4,1216), (52,3), (185,550), (382,5907), (5446,5447), (5562,6101), (5891,2979), (6102,548), (6243,389), (9967,3313), (10263,140), (10575,20)

= anticomplement of X(5446)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,52,9730), (3,568,9729), (3,6243,389), (4,1216,5891), (4,2979,1216), (140,10263,51), (389,6243,52), (394,7387,10539), (578,3098,3), (631,3060,5462), (632,10095,373), (3091,7999,10170), (3523,3567,5892), (3819,10110,1656), (5446,5447,2)

= [ 14.308592523540010, 11.62216907894384, -11.009418352841360 ]

 

Z” = *(2*cos(2*A)+1)*cos(B-C)+3*( cos(A)+cos(3*A))*cos(2*(B-C))+ (-cos(2*A)+cos(4*A)+1)*cos(3*( B-C))-1/2*cos(7*A)+1/2*cos(3* A)+2*cos(5*A) : : (trilinears)

= [ 4.360516333722063, 5.63435438136751, -2.272588397680235 ]

 

César Lozada