[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
The parallels through Na, Nb, Nc to BC, CA, AB, resp. bound a triangle
A*B*C* homothetic to ABC.
Which is the locus of P such that the homothetic center of ABC, A*B*C*
lies on the OP line?
[Angel Montesdeoca]:
*** The locus of P such that the homothetic center of ABC, A*B*C* lies
on the OP line is the McCay cubic.
*** If P=u:v:w (barycentric), then the homothetic center of ABC, A*B*C*
is
Q = v w (-(b^2-c^2)^2 u^2+a^2 u (c^2 (u+v-w)+b^2 (u-v+w))+a^4 (2 v
w+u (v+w))) : ... : ...
*** If Q is the homothetic center of ABC, A*B*C*, pairs {P=X(i), Q=X(j)}, for pairs {i,j }: {1,65}, {3,4}, {4,5}, {6,5640},
{13,5472}, {14,5471}, {54,10095}, {56,14923}, {64,12111}, {84,14872},
{511,13137}, {512,12833}, {523,7471}, {947,10110}.
Q = 4 a^4+a^2 (b^2+c^2)-(b^2-c^2)^2 : ... : ....
Q is the midpoint of X(i) and X(j), for these {i, j}: {6,7737},
{1992,11159}.
Q is the reflection of X(i) in X(j), for these {i, j}: {5,10796},
{141,7804}, {7761,3589}, {14929,141}, {15048,6}.
Q lies on lines X(i)X(j) for these {i, j}: {2,1285}, {3,7736},
{4,3172}, {5,32}, {6,30}, {11,609}, {12,7031}, {20,9605}, {39,550},
{51,15510}, {53,14581}, {69,11286}, {83,7750}, {98,14485}, {99,12156},
{112,251}, {115,3845}, {140,2548}, {141,754}, {172,496}, {183,3793},
{187,549}, {193,14033}, {235,10312}, {315,7819}, {316,7792}, {325,3972},
{376,5024}, {381,7735}, {382,5286}, {384,3933}, {385,8370}, {393,18494},
{428,5359}, {468,9745}, {495,1914}, {524,3734}, {538,3629}, {543,8584},
{546,3767}, {548,5013}, {574,8703}, {597,2030}, {598,3363}, {632,1506},
{952,1572}, {966,11354}, {1003,6390}, {1007,11288}, {1316,6792},
{1353,2782}, {1383,7426}, {1500,10386}, {1503,5039}, {1513,10788},
{1555,8779}, {1595,1968}, {1596,10311}, {1597,3087}, {1609,7514},
{1611,10128}, {1625,3051}, {1657,7738}, {1901,5037}, {1992,11159},
{1995,16317}, {2207,6756}, {2386,9969}, {2393,16983}, {2420,11007},
{2794,5480}, {2896,16988}, {3054,7603}, {3055,11539}, {3058,16785},
{3199,7715}, {3314,6661}, {3329,8356}, {3524,15655}, {3530,5023},
{3541,8778}, {3552,7921}, {3575,8743}, {3589,7761}, {3618,11287},
{3627,5007}, {3820,4386}, {3850,13881}, {3853,5319}, {3858,7755},
{3934,15598}, {5041,7756}, {5103,16385}, {5206,15712}, {5210,12100},
{5276,11113}, {5277,17527}, {5280,6284}, {5299,7354}, {5309,15687},
{5434,16784}, {5471,14137}, {5472,14136}, {5585,14891}, {6423,7584},
{6424,7583}, {6656,7787}, {6658,7839}, {6680,7843}, {6772,9112},
{6775,9113}, {6823,10316}, {7575,9699}, {7576,8744}, {7754,14035},
{7759,7789}, {7766,11361}, {7767,7770}, {7772,15704}, {7773,8361},
{7776,14001}, {7778,8368}, {7784,8364}, {7785,7807}, {7802,7878},
{7803,8357}, {7816,7838}, {7820,7845}, {7829,7842}, {7841,16989},
{7846,7860}, {7873,7889}, {7879,16898}, {7881,14037}, {7885,8363},
{7892,7900}, {7897,14036}, {7929,16895}, {8359,11174}, {8367,15271},
{8573,9818}, {8588,17504}, {8981,12963}, {9465,10301}, {9575,18481},
{9599,15325}, {9698,15513}, {9939,16986}, {10109,18584}, {10317,15760},
{10547,11380}, {11001,14482}, {11163,12040}, {11185,14614},
{11297,11488}, {11298,11489}, {11646,12212}, {11648,14075},
{11842,15980}, {12006,15575}, {12968,13966}, {13357,14881},
{13785,18539}, {15480,17131}, {15603,15700}, {15809,17409},
{16306,18572}.
(6 - 9 - 13) - search numbers of Q: (0.310524995543071,
0.808779387977605, 2.93742106151847).
Angel Montesdeoca
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