[Kadir Altintas and Ercole Suppa]:
Let ABC be a triangle with orthocenter H, P a point and DEF the cevian triangle of P.
Denote:
Na, Nb, Nc = the NPC centers of AFE, FBD, DEC, resp.
La = the line through Na parallel to AH
Define Lb, Lc cyclically
(1) Prove that the lines La, Lb, Lc concur at a point Q
(2) Find the locus of point P such that Q lies on the Euler line
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[Ercole Suppa]:
(1) Let Q = Q(P) the concurrency point of La, Lb, Lc.
*** Pairs {P=X(i),Q=X(j)} : {1,946}, {2,546}, {4,389}, {13,11555}, {14,11556}, {20,6696}, {63,12616}, {75,18480}, {99,3}, {100,18242}, {190,5690}, {514,12699}, {523,3627}, {525,6247}, {648,22802}, {850,18383}, {925,9927}, {3413,3}, {3414,3}, {6189,3627}, {6190,3627}, {6563,22660}, {15412,13419}
*** Some points:
Q(X(3)) = X(4)X(19172) ∩ X(5)X(5944)
= a^12 b^4-5 a^10 b^6+10 a^8 b^8-10 a^6 b^10+5 a^4 b^12-a^2 b^14-2 a^12 b^2 c^2+a^10 b^4 c^2+3 a^8 b^6 c^2+6 a^6 b^8 c^2-16 a^4 b^10 c^2+9 a^2 b^12 c^2-b^14 c^2+a^12 c^4+a^10 b^2 c^4-2 a^8 b^4 c^4+4 a^6 b^6 c^4+11 a^4 b^8 c^4-21 a^2 b^10 c^4+6 b^12 c^4-5 a^10 c^6+3 a^8 b^2 c^6+4 a^6 b^4 c^6+13 a^2 b^8 c^6-15 b^10 c^6+10 a^8 c^8+6 a^6 b^2 c^8+11 a^4 b^4 c^8+13 a^2 b^6 c^8+20 b^8 c^8-10 a^6 c^10-16 a^4 b^2 c^10-21 a^2 b^4 c^10-15 b^6 c^10+5 a^4 c^12+9 a^2 b^2 c^12+6 b^4 c^12-a^2 c^14-b^2 c^14 : : (barys)
= S^4 + (16 R^4-2 R^2 SB-2 R^2 SC+SB SC-8 R^2 SW+SW^2)S^2 -16 R^4 SB SC+4 R^2 SB SC SW+SB SC SW^2 : : (barys)
= lies on these lines: {4,19172}, {5,5944}, {389,7668}, {523,14978}, {578,3613}, {1510,18488}, {3574,8901}, {7404,15270}
= (6-9-13) search numbers [-0.474335339576601546, 34.4196101560102575, -19.9693723924489061]
Q(X(5)) = X(5)X(27684) ∩ X(2883)X(3845)
= 4 a^16-22 a^14 b^2+45 a^12 b^4-31 a^10 b^6-30 a^8 b^8+76 a^6 b^10-63 a^4 b^12+25 a^2 b^14-4 b^16-22 a^14 c^2+58 a^12 b^2 c^2-27 a^10 b^4 c^2-21 a^8 b^6 c^2-56 a^6 b^8 c^2+156 a^4 b^10 c^2-119 a^2 b^12 c^2+31 b^14 c^2+45 a^12 c^4-27 a^10 b^2 c^4-18 a^8 b^4 c^4-20 a^6 b^6 c^4-81 a^4 b^8 c^4+207 a^2 b^10 c^4-106 b^12 c^4-31 a^10 c^6-21 a^8 b^2 c^6-20 a^6 b^4 c^6-24 a^4 b^6 c^6-113 a^2 b^8 c^6+209 b^10 c^6-30 a^8 c^8-56 a^6 b^2 c^8-81 a^4 b^4 c^8-113 a^2 b^6 c^8-260 b^8 c^8+76 a^6 c^10+156 a^4 b^2 c^10+207 a^2 b^4 c^10+209 b^6 c^10-63 a^4 c^12-119 a^2 b^2 c^12-106 b^4 c^12+25 a^2 c^14+31 b^2 c^14-4 c^16 : : (barys)
= 3 S^4 + (-16 R^4-10 R^2 SB-10 R^2 SC+27 SB SC+8 R^2 SW+4 SB SW+4 SC SW-SW^2)S^2 +16 R^4 SB SC-12 R^2 SB SC SW-SB SC SW^2 : : (barys)
= lies on these lines: {5,27684}, {2883,3845}
= (6-9-13) search numbers [17.0430850547568364, 19.3009349735343774, -17.5875605250425759]
Q(X(6)) = REFLECTION OF X(14134) IN X(140)
= 3 a^6 b^4-2 a^4 b^6-a^2 b^8+2 a^6 b^2 c^2+6 a^4 b^4 c^2+a^2 b^6 c^2-b^8 c^2+3 a^6 c^4+6 a^4 b^2 c^4+b^6 c^4-2 a^4 c^6+a^2 b^2 c^6+b^4 c^6-a^2 c^8-b^2 c^8 : : (barys)
= (2 R^2+SB+SC)S^4 + (-14 R^2 SB SC+2 SB SC SW+SB SW^2+SC SW^2)S^2 2 SB SC SW^3 : : (barys)
= 2*X[140]-X[14134]
= lies on these lines: {5,6310}, {30,14133}, {140,14134}, {575,3627}
= reflection of X(14134) in X(140)
= (6-9-13) search numbers [-1.55581365296246496, -1.68069337784159518, 5.52228927562662113]
Q(X(7))= MIDPOINT OF X(4) AND X(5884)
= a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-2 a^4 b c+a^3 b^2 c-a^2 b^3 c-2 a b^4 c+3 b^5 c-a^4 c^2+a^3 b c^2-2 a^2 b^2 c^2+a b^3 c^2+b^4 c^2-2 a^3 c^3-a^2 b c^3+a b^2 c^3-6 b^3 c^3+2 a^2 c^4-2 a b c^4+b^2 c^4+a c^5+3 b c^5-c^6) : : (barys)
= 2 R S^3 + (12 a R^2-12 b R^2+2 a SB+2 b SB-c SB+2 a SC-b SC+2 c SC-4 a SW+3 b SW)S^2 + 10 R S SB SC+3 b SB SC^2-3 c SB SC^2-3 b SB SC SW : : (barys)
= X[3]-3*X[5883], X[20]-5*X[15016], X[72]-3*X[10175], 3*X[354]-X[5882], X[355]+X[3874], X[944]-5*X[18398], 5*X[1656]-3*X[10176], 7*X[3090]-3*X[5692], 5*X[3091]-X[5693], X[3678]-2*X[9956], 3*X[3753]-X[11362], 3*X[3817]+X[4084], X[3868]+3*X[5587], X[3869]-5*X[8227], 3*X[3873]+X[5881], 3*X[3877]-7*X[9624], X[3878]-3*X[5886], X[3901]+7*X[7989], 3*X[3919]+X[4301], X[4297]-3*X[10202], X[4757]+2*X[9955], 2*X[5044]-3*X[10172], 2*X[5045]-X[13607], 5*X[5439]-3*X[10165], 5*X[5818]-X[5904], X[12528]-5*X[18492], X[14872]+3*X[24473], X[14923]+3*X[16200]
= lies on these lines: {1,1389}, {3,5883}, {4,79}, {5,758}, {7,6256}, {10,6881}, {11,65}, {20,15016}, {21,5535}, {30,5885}, {40,1621}, {57,5450}, {72,10175}, {104,3337}, {140,517}, {191,6920}, {354,5882}, {355,3874}, {389,2779}, {499,1788}, {515,942}, {546,2771}, {912,19925}, {944,18398}, {950,13750}, {952,3881}, {958,2095}, {971,16616}, {1012,5221}, {1158,3339}, {1376,1482}, {1476,3333}, {1512,13407}, {1532,3649}, {1656,10176}, {1735,2654}, {1737,15556}, {1768,21669}, {1837,18389}, {1845,1940}, {2392,5446}, {2778,6696}, {2801,18480}, {2802,10222}, {2829,24470}, {3075,11700}, {3090,5692}, {3091,5693}, {3109,18180}, {3336,6906}, {3585,6246}, {3647,7489}, {3656,10199}, {3671,7682}, {3678,9956}, {3753,11362}, {3817,4084}, {3826,3918}, {3868,5587}, {3869,8227}, {3873,5881}, {3877,9624}, {3878,5886}, {3901,7989}, {3919,4301}, {4292,18838}, {4295,5804}, {4297,10202}, {4757,9955}, {5044,10172}, {5045,13607}, {5218,5697}, {5330,7982}, {5425,21740}, {5439,10165}, {5538,6940}, {5563,11715}, {5570,10106}, {5708,12114}, {5720,12559}, {5734,26062}, {5755,25081}, {5761,26364}, {5806,5893} {5818,5904}, {5836,28234}, {5842,12433}, {6147,18242}, {6261,11529}, {6326,6915}, {6842,11263}, {6853,26725}, {6894,9803}, {6911,22836}, {6918,12635}, {7680,12432}, {8070,12047}, {9943,28150}, {10044,10051}, {10532,10573}, {10597,12647}, {10980,12650}, {11009,25485}, {11246,18977}, {11500,15934}, {11520,17857}, {12245,26040}, {12528,18492}, {12704,19860}, {13145,28174}, {13369,28164}, {13729,14450}, {14872,24473}, {14923,16200}, {18240,24928}, {18391,26332}, {26201,28186}
= midpoint of X(i) and X(j) for these {i,j}: {4,5884}, {10,24474}, {65,946}, {355,3874}, {942,7686}, {4084,5887}, {6246,11570}, {18480,24475}
= reflection of X(i) in X(j) for these {i,j}: {3678,9956}, {3881,6583}, {3884,5901}, {5690,3918}, {6684,3812}, {12005,942}, {13464,13374}, {13607,5045}, {18483,5806}, {20117,5}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,5902,5884}, {946,10265,6831}, {3671,7682,12608}, {3817,4084,5887}, {4295,5804,26333}, {5439,14110,10165}
= (6-9-13) search numbers [-0.731714568614849235, -0.844221214062683813, 4.56283973869616754]
Q(X(8)) = MIDPOINT OF X(10) AND X(12688)
= a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c+2 a^4 b c+a^3 b^2 c+a^2 b^3 c-2 a b^4 c-3 b^5 c-a^4 c^2+a^3 b c^2-2 a^2 b^2 c^2+a b^3 c^2+b^4 c^2-2 a^3 c^3+a^2 b c^3+a b^2 c^3+6 b^3 c^3+2 a^2 c^4-2 a b c^4+b^2 c^4+a c^5-3 b c^5-c^6) : : (barys)
= 2 R S^3 - (12 a R^2-12 b R^2+2 a SB+2 b SB-c SB+2 a SC-b SC+2 c SC-4 a SW+3 b SW)S^2 +10 R S SB SC-3 b SB SC^2+3 c SB SC^2+3 b SB SC SW : : (barys)
= 4*X[5]-3*X[3833], X[10]-3*X[5927], X[20]-3*X[10176], X[40]-2*X[4015], X[185]-3*X[15049], 3*X[210]-X[5493], 3*X[381]-X[5884], 3*X[551]-X[12680], X[942]-2*X[12571], X[1071]-3*X[3817], 5*X[1698]-X[9961], 3*X[1699]-X[3874], 5*X[3091]-3*X[5883], X[3146]+3*X[5692], 9*X[3545]-5*X[15016], 7*X[3624]-3*X[11220], X[3626]-2*X[9947], X[3627]+X[5694], 2*X[3634]-X[9943], 3*X[3681]+X[9589], 7*X[3832]-3*X[5902], 2*X[3850]-X[5885], X[3878]+X[5691], 3*X[3892]-5*X[11522], 2*X[3918]-3*X[5587], 3*X[4134]-X[7957], X[4301]+X[14872], 4*X[4540]-3*X[5657], 4*X[4547]-X[6361], X[4757]-2*X[7686], 2*X[5044]-X[12512], X[5904]+3*X[9812], 9*X[9779]-5*X[18398], 2*X[9940]-3*X[10171], 2*X[9955]-X[12005], 3*X[10167]-5*X[19862], 3*X[11227]-4*X[19878], X[17661]+X[21630]
= lies on these lines: {2,16120}, {3,16112}, {4,758}, {5,3833}, {9,12511}, {10,5927}, {20,10176}, {30,20117}, {40,4015}, {58,9355}, {79,6894}, {185,15049}, {210,5493}, {226,1898}, {381,5884}, {389,2772}, {411,3647}, {515,3884}, {516,3678}, {517,3853}, {519,9856}, {546,2771}, {551,12680}, {912,18483}, {936,3062}, {942,12571}, {946,2801}, {952,26200}, {960,28164}, {971,1125}, {991,27784}, {1071,3817}, {1490,5248}, {1698,9961}, {1699,3874}, {1709,25440}, {1750,12514}, {1768,6915}, {1836,12432}, {1864,3671}, {2392,5907}, {2800,18480}, {2802,12672}, {3091,5883}, {3146,5692}, {3159,28850}, {3244,13227}, {3337,13243}, {3434,12059}, {3545,15016}, {3624,11220}, {3626,9947}, {3627,5694}, {3634,9943}, {3681,9589}, {3754,6001}, {3811,11372}, {3822,6260}, {3825,6245}, {3832,5902}, {3850,5885}, {3878,5691}, {3892,11522}, {3918,5587}, {3947,12711}, {4134,7957}, {4187,17653}, {4301,14872}, {4540,5657}, {4547,6361}, {4757,7686}, {4882,11678}, {5044,12512}, {5658,10198}, {5851,24470}, {5904,9812}, {6147,20116}, {6261,18540}, {6326,21669}, {6681,6705}, {6702,12616}, {6826,16127}, {6831,21635}, {6905,7701}, {6920,16132}, {6940,10308}, {6986,16143}, {8226,11263}, {8581,21625}, {8715,12705}, {9779,18398}, {9842,9948}, {9940,10171}, {9955,12005}, {10122,10883}, {10167,19862}, {11227,19878}, {12609,12664}, {12684,25524}, {12691,18406}, {17661,21630}, {18250,18251}, {21077,21628}
= midpoint of X(i) and X(j) for these {i,j}: {10,12688}, {3627,5694}, {3874,12528}, {3878,5691}, {4301,14872}, {9949,17646}, {17661,21630}
= reflection of X(i) in X(j) for these {i,j}: {40,4015}, {942,12571}, {3626,9947}, {3678,5777}, {3754,19925}, {3881,946}, {4757,7686}, {5885,3850}, {9943,3634}, {12005,9955}, {12512,5044}, {12564,12558}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {40,15064,4015}, {1699,12528,3874}, {3091,15071,5883}, {5927,12688,10}, {6260,12617,3822}, {6894,9809,79}, {9943,10157,3634}
= (6-9-13) search numbers [-4.39203310067678252, -6.00504972529966928, 9.82509879973497285]
Q(X(10)) = X(4)X(3617) ∩ X(381)X(19722)
= 2 a^7+5 a^6 b+3 a^5 b^2+a^4 b^3+a^3 b^4-4 a^2 b^5-6 a b^6-2 b^7+5 a^6 c+8 a^5 b c+4 a^4 b^2 c+4 a^3 b^3 c-3 a^2 b^4 c-12 a b^5 c-6 b^6 c+3 a^5 c^2+4 a^4 b c^2+6 a^3 b^2 c^2+13 a^2 b^3 c^2+6 a b^4 c^2-2 b^5 c^2+a^4 c^3+4 a^3 b c^3+13 a^2 b^2 c^3+24 a b^3 c^3+10 b^4 c^3+a^3 c^4-3 a^2 b c^4+6 a b^2 c^4+10 b^3 c^4-4 a^2 c^5-12 a b c^5-2 b^2 c^5-6 a c^6-6 b c^6-2 c^7 : : (barys)
= 8 R S^3 - (28 a R^2-28 b R^2+3 a SB+b SB+2 c SB+3 a SC+2 b SC+c SC-11 a SW+4 b SW-3 c SW)S^2 + 40 R S SB SC-7 b SB SC^2+7 c SB SC^2+13 a SB SC SW+20 b SB SC SW+13 c SB SC SW : : (barys)
= lies on these lines X(i)X(j) for these {i,j}: {4,3617}, {381,19722}
= (6-9-13) search numbers [-2.92404697621112577, -3.99834100688876336, 7.75830686031250409]
(2) The locus of point P such that Q lies on the Euler line is the cubic K242
K242: ∑ (a-c) (b-c) (a+c) (b+c) y^2 z+(a-b) (a+b) (b-c) (b+c) y z^2 = 0
X(i) lies on K242 for these i: {2,99,523,1113,1114,3413,3414,6189,6190,22339,22340}
Pairs {P=X(i) on K242, Q=X(j)} : {2,546}, {99,3}, {523,3627}, {3413,3},{3414,3}, {6189,3627}, {6190,3627}
Best regards
Ercole Suppa
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