HYACINTHOS 29094

[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.

MaMbMc  = the medial triangle of NaNbNc

M1, M2, M3 = the reflections of Ma, Mb, Mc in BC, CA, AB, resp.

Which is the locus of P such that A'B'C', M1M2M3 are orthologic?
N lies on the locus.


[Peter Moses]:

Hi Antreas,

Locus Inf + circular circum-quintic through {3,4,5,1113,1114,25148}.

Orthologies

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P= X(3)

(A'B'C',  M1M2M3)

 

= (a^10-3 a^8 b^2+a^6 b^4+5 a^4 b^6-6 a^2 b^8+2 b^10-3 a^8 c^2+7 a^6 b^2 c^2-5 a^4 b^4 c^2+7 a^2 b^6 c^2-6 b^8 c^2+a^6 c^4-5 a^4 b^2 c^4-2 a^2 b^4 c^4+4 b^6 c^4+5 a^4 c^6+7 a^2 b^2 c^6+4 b^4 c^6-6 a^2 c^8-6 b^2 c^8+2 c^10) (2 a^10 b^2-6 a^8 b^4+4 a^6 b^6+4 a^4 b^8-6 a^2 b^10+2 b^12+2 a^10 c^2-12 a^8 b^2 c^2+12 a^6 b^4 c^2-a^4 b^6 c^2+4 a^2 b^8 c^2-5 b^10 c^2-6 a^8 c^4+12 a^6 b^2 c^4-10 a^4 b^4 c^4+2 a^2 b^6 c^4+2 b^8 c^4+4 a^6 c^6-a^4 b^2 c^6+2 a^2 b^4 c^6+2 b^6 c^6+4 a^4 c^8+4 a^2 b^2 c^8+2 b^4 c^8-6 a^2 c^10-5 b^2 c^10+2 c^12) : :

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(M1M2M3,  A'B'C') = COMPLEMENT OF X(10282)

 

=  a^8*b^2 - a^6*b^4 - 3*a^4*b^6 + 5*a^2*b^8 - 2*b^10 + a^8*c^2 - 4*a^6*b^2*c^2 + 3*a^4*b^4*c^2 - 6*a^2*b^6*c^2 + 6*b^8*c^2 - a^6*c^4 + 3*a^4*b^2*c^4 + 2*a^2*b^4*c^4 - 4*b^6*c^4 - 3*a^4*c^6 - 6*a^2*b^2*c^6 - 4*b^4*c^6 + 5*a^2*c^8 + 6*b^2*c^8 - 2*c^10 : :

= 9 X[2] - X[9833],3 X[2] + X[18381],5 X[3] + 3 X[18405],X[3] + 3 X[23325],X[4] + 3 X[23329],5 X[5] - X[2883],3 X[5] + X[6247],X[5] + 3 X[23332],X[20] + 3 X[18376],X[64] + 7 X[3851],3 X[154] - 11 X[5070],3 X[381] + X[3357],9 X[381] - X[5895],X[382] + 3 X[11204],3 X[547] - X[16252],X[550] - 3 X[10193],X[550] + 3 X[23324],5 X[632] - 3 X[10182],X[1147] - 5 X[31283],X[1498] - 9 X[5055],5 X[1656] + 3 X[1853],5 X[1656] - X[6759],5 X[1656] + X[14864],3 X[1853] + X[6759],3 X[1853] - X[14864],3 X[2883] + 5 X[6247],X[2883] + 5 X[20299],X[2883] + 15 X[23332],7 X[3090] + X[14216],15 X[3091] + X[12250],5 X[3091] - X[22802],3 X[3357] + X[5895],7 X[3526] - 3 X[11202],9 X[3545] - X[5878],X[3627] + 3 X[23328],3 X[3830] + 5 X[8567],7 X[3832] + X[20427],5 X[3843] + 3 X[10606],3 X[3845] + X[5894],9 X[5054] - X[17845],3 X[5066] - X[5893],13 X[5067] + 3 X[32064],15 X[5071] + X[12324],3 X[5656] - 19 X[15022],X[6247] - 3 X[20299],X[6247] - 9 X[23332],X[8549] + 3 X[11178],X[9833] - 3 X[10282],X[9833] + 3 X[18381],X[9927] + 3 X[18281],3 X[10250] + X[15069],X[12250] + 3 X[22802],X[13093] + 15 X[19709],X[13289] - 5 X[15059],X[13293] + 3 X[14644],X[13346] + 3 X[14852],3 X[14076] - X[21230],5 X[14530] - 21 X[15703],5 X[15027] + 3 X[15131],7 X[15057] - 3 X[16219],3 X[15061] + X[19506],5 X[18383] - 3 X[18405],X[18383] - 3 X[23325],X[18405] - 5 X[23325],X[20299] - 3 X[23332],X[21230] + 3 X[32351]  

= lies on these lines: {2, 9833}, {3, 18383}, {4, 11270}, {5, 2883}, {20, 18376}, {30, 20191}, {51, 26917}, {64, 3851}, {125, 389}, {140, 13470}, {154, 5070}, {185, 7577}, {186, 11572}, {343, 15606}, {381, 3357}, {382, 11204}, {403, 13474}, {427, 10110}, {468, 13419}, {511, 5449}, {542, 9820}, {546, 2777}, {547, 16252}, {550, 10193}, {575, 18952}, {578, 5094}, {632, 10182}, {858, 15644}, {1092, 30744}, {1147, 31283}, {1204, 7547}, {1209, 3819}, {1216, 14076}, {1493, 11232}, {1495, 14940}, {1498, 5055}, {1503, 3628}, {1531, 11440}, {1656, 1853}, {1885, 7687}, {2072, 5907}, {2781, 10095}, {3090, 14216}, {3091, 7703}, {3520, 13851}, {3526, 11202}, {3541, 18390}, {3545, 5878}, {3581, 32395}, {3627, 23328}, {3830, 8567}, {3832, 20427}, {3843, 10606}, {3845, 5894}, {3850, 15311}, {5020, 32321}, {5054, 17845}, {5066, 5893}, {5067, 32064}, {5071, 12324}, {5498, 30522}, {5562, 23293}, {5576, 5943}, {5651, 31282}, {5656, 15022}, {5663, 32184}, {5972, 12134}, {6143, 12254}, {6640, 18474}, {6723, 16238}, {7505, 11550}, {7507, 11438}, {7553, 32223}, {7574, 32365}, {7603, 32445}, {8549, 11178}, {8681, 23307}, {9927, 15123}, {10113, 25564}, {10117, 18369}, {10224, 13561}, {10226, 18379}, {10250, 15069}, {10254, 10575}, {10255, 12162}, {10274, 13353}, {11245, 12242}, {11264, 32376}, {11381, 16868}, {11585, 11793}, {12006, 13413}, {12370, 15113}, {13093, 19709}, {13154, 15577}, {13160, 16836}, {13289, 15059}, {13293, 14644}, {13346, 14852}, {13382, 18388}, {13383, 29012}, {13406, 14915}, {14530, 15703}, {14627, 17847}, {14810, 18382}, {15027, 15131}, {15037, 17824}, {15057, 16219}, {15061, 19506}, {15116, 20301}, {15578, 17714}, {17702, 23336}, {17712, 25337}, {18488, 23515}, {18567, 32210}, {23300, 24206}, {31724, 32110}

= complement of X(10282)

 

= complement of the isogonal of X(15319) 

 

= X(15319)-complementary conjugate of X(10)

 
= midpoint of X(i) and X(j) for these {i,j}: {3, 18383}, {5, 20299}, {125, 32743}, {546, 6696}, {5449, 13371}, {6697, 20300}, {6759, 14864}, {10113, 25564}, {10193, 23324}, {10224, 13561}, {10226, 18379}, {10282, 18381}, {14076, 32351}, {14810, 18382}, {15116, 20301}, {18567, 32210}, {23300, 24206}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 18381, 10282}, {3, 23325, 18383}, {5, 23332, 20299}, {125, 1594, 389}, {125, 3574, 26879}, {1594, 26879, 3574}, {1656, 1853, 6759}, {1853, 6759, 14864}, {3574, 26879, 389}, {6143, 25739, 13367}, {7577, 23294, 185}, {11585, 21243, 11793}, {13353, 15139, 10274}

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P = X(4)

(A'B'C',  M1M2M3) = X(6152)

(M1M2M3,  A'B'C') = X(52)

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P = X(5)

(A'B'C', M1M2M3) =

 

= a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^14 - 9*a^12*b^2 + 31*a^10*b^4 - 55*a^8*b^6 + 55*a^6*b^8 - 31*a^4*b^10 + 9*a^2*b^12 - b^14 - 9*a^12*c^2 + 48*a^10*b^2*c^2 - 87*a^8*b^4*c^2 + 56*a^6*b^6*c^2 + 9*a^4*b^8*c^2 - 24*a^2*b^10*c^2 + 7*b^12*c^2 + 31*a^10*c^4 - 87*a^8*b^2*c^4 + 51*a^6*b^4*c^4 + 13*a^4*b^6*c^4 + 7*a^2*b^8*c^4 - 15*b^10*c^4 - 55*a^8*c^6 + 56*a^6*b^2*c^6 + 13*a^4*b^4*c^6 + 16*a^2*b^6*c^6 + 9*b^8*c^6 + 55*a^6*c^8 + 9*a^4*b^2*c^8 + 7*a^2*b^4*c^8 + 9*b^6*c^8 - 31*a^4*c^10 - 24*a^2*b^2*c^10 - 15*b^4*c^10 + 9*a^2*c^12 + 7*b^2*c^12 - c^14)*(a^20 - 10*a^18*b^2 + 39*a^16*b^4 - 76*a^14*b^6 + 70*a^12*b^8 - 70*a^8*b^12 + 76*a^6*b^14 - 39*a^4*b^16 + 10*a^2*b^18 - b^20 - 10*a^18*c^2 + 57*a^16*b^2*c^2 - 120*a^14*b^4*c^2 + 110*a^12*b^6*c^2 - 56*a^10*b^8*c^2 + 100*a^8*b^10*c^2 - 192*a^6*b^12*c^2 + 170*a^4*b^14*c^2 - 70*a^2*b^16*c^2 + 11*b^18*c^2 + 39*a^16*c^4 - 120*a^14*b^2*c^4 + 119*a^12*b^4*c^4 - 58*a^10*b^6*c^4 + 32*a^8*b^8*c^4 + 98*a^6*b^10*c^4 - 263*a^4*b^12*c^4 + 208*a^2*b^14*c^4 - 55*b^16*c^4 - 76*a^14*c^6 + 110*a^12*b^2*c^6 - 58*a^10*b^4*c^6 + 29*a^8*b^6*c^6 + 18*a^6*b^8*c^6 + 121*a^4*b^10*c^6 - 304*a^2*b^12*c^6 + 160*b^14*c^6 + 70*a^12*c^8 - 56*a^10*b^2*c^8 + 32*a^8*b^4*c^8 + 18*a^6*b^6*c^8 + 22*a^4*b^8*c^8 + 156*a^2*b^10*c^8 - 296*b^12*c^8 + 100*a^8*b^2*c^10 + 98*a^6*b^4*c^10 + 121*a^4*b^6*c^10 + 156*a^2*b^8*c^10 + 362*b^10*c^10 - 70*a^8*c^12 - 192*a^6*b^2*c^12 - 263*a^4*b^4*c^12 - 304*a^2*b^6*c^12 - 296*b^8*c^12 + 76*a^6*c^14 + 170*a^4*b^2*c^14 + 208*a^2*b^4*c^14 + 160*b^6*c^14 - 39*a^4*c^16 - 70*a^2*b^2*c^16 - 55*b^4*c^16 + 10*a^2*c^18 + 11*b^2*c^18 - c^20) : :

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(M1M2M3,  A'B'C') =

 

= a^2*(a^24*b^2 - 12*a^22*b^4 + 60*a^20*b^6 - 164*a^18*b^8 + 261*a^16*b^10 - 216*a^14*b^12 + 216*a^10*b^16 - 261*a^8*b^18 + 164*a^6*b^20 - 60*a^4*b^22 + 12*a^2*b^24 - b^26 + a^24*c^2 - 22*a^22*b^2*c^2 + 137*a^20*b^4*c^2 - 403*a^18*b^6*c^2 + 637*a^16*b^8*c^2 - 536*a^14*b^10*c^2 + 238*a^12*b^12*c^2 - 218*a^10*b^14*c^2 + 463*a^8*b^16*c^2 - 522*a^6*b^18*c^2 + 305*a^4*b^20*c^2 - 91*a^2*b^22*c^2 + 11*b^24*c^2 - 12*a^22*c^4 + 137*a^20*b^2*c^4 - 512*a^18*b^4*c^4 + 873*a^16*b^6*c^4 - 706*a^14*b^8*c^4 + 251*a^12*b^10*c^4 - 68*a^10*b^12*c^4 - 123*a^8*b^14*c^4 + 522*a^6*b^16*c^4 - 604*a^4*b^18*c^4 + 296*a^2*b^20*c^4 - 54*b^22*c^4 + 60*a^20*c^6 - 403*a^18*b^2*c^6 + 873*a^16*b^4*c^6 - 774*a^14*b^6*c^6 + 294*a^12*b^8*c^6 - 79*a^10*b^10*c^6 - 52*a^8*b^12*c^6 - 63*a^6*b^14*c^6 + 499*a^4*b^16*c^6 - 509*a^2*b^18*c^6 + 154*b^20*c^6 - 164*a^18*c^8 + 637*a^16*b^2*c^8 - 706*a^14*b^4*c^8 + 294*a^12*b^6*c^8 - 80*a^10*b^8*c^8 - 27*a^8*b^10*c^8 - 80*a^6*b^12*c^8 - 23*a^4*b^14*c^8 + 424*a^2*b^16*c^8 - 275*b^18*c^8 + 261*a^16*c^10 - 536*a^14*b^2*c^10 + 251*a^12*b^4*c^10 - 79*a^10*b^6*c^10 - 27*a^8*b^8*c^10 - 42*a^6*b^10*c^10 - 117*a^4*b^12*c^10 - 8*a^2*b^14*c^10 + 297*b^16*c^10 - 216*a^14*c^12 + 238*a^12*b^2*c^12 - 68*a^10*b^4*c^12 - 52*a^8*b^6*c^12 - 80*a^6*b^8*c^12 - 117*a^4*b^10*c^12 - 248*a^2*b^12*c^12 - 132*b^14*c^12 - 218*a^10*b^2*c^14 - 123*a^8*b^4*c^14 - 63*a^6*b^6*c^14 - 23*a^4*b^8*c^14 - 8*a^2*b^10*c^14 - 132*b^12*c^14 + 216*a^10*c^16 + 463*a^8*b^2*c^16 + 522*a^6*b^4*c^16 + 499*a^4*b^6*c^16 + 424*a^2*b^8*c^16 + 297*b^10*c^16 - 261*a^8*c^18 - 522*a^6*b^2*c^18 - 604*a^4*b^4*c^18 - 509*a^2*b^6*c^18 - 275*b^8*c^18 + 164*a^6*c^20 + 305*a^4*b^2*c^20 + 296*a^2*b^4*c^20 + 154*b^6*c^20 - 60*a^4*c^22 - 91*a^2*b^2*c^22 - 54*b^4*c^22 + 12*a^2*c^24 + 11*b^2*c^24 - c^26) : :
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Best regards,
Peter Moses.