HYACINTHOS 27487

[ Tran Quang Hung]:

Let ABC be a triangle.

P is a point on Euler line.

A'B'C' is pedal triangle of P.

Na,Nb,Nc are NPC centers of PBC,PCA,PAB, resp.

Then triangles A'B'C' and NaNbNc are perspective.

The perspector always lies on a parabola ? Which is it ?

 

[Angel Montesdeoca]:

**** The triangle A'B'C' and NaNbNc are perspective, the perspector Q
lies on the hyperbola that passes through points 3, 5, 5663, 10539,
13394.

One point of infinity is X( 5663) and the other is:

2 a^10+a^2 (b^2-c^2)^4-4 a^8 (b^2+c^2)-(b^2-c^2)^4 (b^2+c^2)+a^6 (b^4+8
b^2 c^2+c^4)+a^4 (b^6-3 b^4 c^2-3 b^2 c^4+c^6),-a^10+a^8 (b^2+3 c^2)+a^6
(b^4-4 b^2 c^2-2 c^4)+(b^2-c^2)^3 (2 b^4+2 b^2 c^2+c^4)+a^4 (b^6-3 b^4
c^2+6 b^2 c^4-2 c^6)+a^2 (-4 b^8+8 b^6 c^2-3 b^4 c^4-4 b^2 c^6+3 c^8 :
.... : ...

with (6 - 9 - 13) - search numbers (1.22023184857025,
1.04347413196238, -1.28558909839099).

Lies on these (parallel) lines:
{1,12888}, {2,14644}, {3,125}, {4,110}, {5,1511}, {6,7706}, {15,10663},
{16,10664}, {20,68}, {26,9938}, {30,511}, {40,12407}, {51,16222},
{52,1986}, {55,12903}, {56,12904}, {67,1350}, {69,4549}, {99,11005},
{104,10778}, {140,6723}, {143,11561}, {146,3146}, {155,382},
{182,15118}, {185,10111}, {195,14049}, {323,10296}, {355,12778},
{371,12891}, {372,12892}, {376,9140}, {378,12827}, {381,5642},
{389,11800}, {477,15396}, {485,8998}, {486,10820}, {546,9820},
{548,13470}, {550,10264}, {576,12596}, {578,12228}, {631,15051},
{671,11656}, {858,10564}, {895,4846}, {944,7984}, {946,11720},
{962,7978}, {974,6146}, {1069,12374}, {1112,3575}, {1141,15958},
{1151,8994}, {1152,13969}, {1160,12804}, {1161,12803}, {1209,14118},
{1216,12358}, {1351,5095}, {1352,4550}, {1385,11735}, {1478,10088},
{1479,10091}, {1482,12898}, {1495,11799}, {1498,11744}, {1531,3292},
{1539,3627}, {1550,7472}, {1593,12168}, {1614,3047}, {1656,15040},
{1657,10620}, {1885,12133}, {2072,13851}, {2079,15538}, {2420,6103},
{2930,14982}, {2935,9937}, {2948,5691}, {3024,6284}, {3028,7354},
{3090,15020}, {3091,15034}, {3157,12373}, {3167,3830}, {3258,14934},
{3398,12201}, {3519,11559}, {3523,15036}, {3525,15025}, {3528,15057},
{3529,11411}, {3534,15041}, {3543,9143}, {3545,11693}, {3574,11597},
{3580,10295}, {3628,15088}, {3818,12584}, {3850,13392}, {4297,11709},
{4299,10071}, {4302,10055}, {5066,11694}, {5073,12164}, {5447,12362},
{5462,9826}, {5465,9880}, {5480,6593}, {5512,9129}, {5562,7723},
{5870,7733}, {5871,7732}, {5889,7722}, {5892,12099}, {6036,15359},
{6102,10112}, {6152,11560}, {6237,7724}, {6238,7727}, {6241,12284},
{6243,14448}, {6247,11598}, {6288,14130}, {6321,16278}, {6391,10293},
{6592,10213}, {6689,13160}, {6841,16164}, {7352,10483}, {7387,10117},
{7471,15468}, {7553,16105}, {7574,15136}, {7576,12824}, {7725,9929},
{7726,9930}, {7731,15102}, {7748,14901}, {7749,8571}, {8548,9976},
{8717,11579}, {8981,13915}, {9181,16188}, {9730,12022}, {9818,10516},
{9821,12501}, {9825,15465}, {9833,9934}, {9834,13208}, {9835,13209},
{9838,13215}, {9839,13216}, {9873,13210}, {9896,9904}, {9908,9919},
{9923,9984}, {9932,12084 },{10024,13367}, {10201,11202}, {10226,13561},
{10282,15761}, {10297,11064}, {10525,12889}, {10526,12890},
{10610,11804}, {10619,15089}, {10625,12225}, {10657,10661},
{10658,10662}, {10665,12375}, {10666,12376}, {10669,12894},
{10673,12895} ,{10679,12905} , {10680,12906},
{10693,14110},{10723,15342},{ 10724,10767},{10750,14499},{ 10751,14500},{10752,11061},{ 11006,12117},
{11248,12334}, {11251,12790}, {11252,12466}, {11253,12467},
{11412,12219}, {11414,13171}, {11449,16868}, {11500,13204},
{11566,12002}, {11807,11819}, {12042,15535}, {12110,13193},
{12111,12273}, {12112,12364}, {12113,13212}, {12114,13213},
{12115,13217}, {12116,13218}, {12160,12165}, {12161,12227},
{12162,12292}, {12192,12193}, {12327,12328}, {12365,12415},
{12366,12416}, {12369,12418}, {12371,12422}, {12372,12423},
{12377,12426}, {12378,12427}, {12381,12430}, {12382,12431},
{13201,15100}, {13292,13568}, {13346,15132}, {13358,13630},
{13394,15760}, {13966,13979}, {14561,15462}, {14677,15704},
{15122,15123}.

Barycentric equation of hyperbola:

-b^2 (b-c)^2 c^2 (b+c)^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (3 a^8-7 a^6 b^2+3
a^4 b^4+3 a^2 b^6-2 b^8-7 a^6 c^2+13 a^4 b^2 c^2-7 a^2 b^4 c^2+b^6 c^2+3
a^4 c^4-7 a^2 b^2 c^4+2 b^4 c^4+3 a^2 c^6+b^2 c^6-2 c^8)x^2+
a^2 (a-b) (a+b) (a-c) (a+c) (a^2-b^2-c^2) (2 a^10 b^2-7 a^8 b^4+8 a^6
b^6-2 a^4 b^8-2 a^2 b^10+b^12+2 a^10 c^2+6 a^8 b^2 c^2-6 a^6 b^4 c^2-10
a^4 b^6 c^2+4 a^2 b^8 c^2+4 b^10 c^2-7 a^8 c^4-6 a^6 b^2 c^4+24 a^4 b^4
c^4-2 a^2 b^6 c^4-25 b^8 c^4+8 a^6 c^6-10 a^4 b^2 c^6-2 a^2 b^4 c^6+40
b^6 c^6-2 a^4 c^8+4 a^2 b^2 c^8-25 b^4 c^8-2 a^2 c^10+4 b^2 c^10+c^12)y
z + .... = 0.

*** The envelope of the lines PQ is the parabola of focus X(1304) and
directrix X(107)X(110). This parabola passes through points X(4),
X(2777) on the line at infinity , X(6225), X(6759).

Equation parabola:

(b-c)^2 (b+c)^2 (a^2-b^2-c^2)^2 (a^8 b^4-4 a^6 b^6+6 a^4 b^8-4 a^2
b^10+b^12+14 a^8 b^2 c^2-12 a^6 b^4 c^2-16 a^4 b^6 c^2+12 a^2 b^8 c^2+2
b^10 c^2+a^8 c^4-12 a^6 b^2 c^4+36 a^4 b^4 c^4-8 a^2 b^6 c^4-17 b^8
c^4-4 a^6 c^6-16 a^4 b^2 c^6-8 a^2 b^4 c^6+28 b^6 c^6+6 a^4 c^8+12 a^2
b^2 c^8-17 b^4 c^8-4 a^2 c^10+2 b^2 c^10+c^12)x^2
-2 (a-b) (a+b) (a-c) (a+c) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^12-a^10 b^2-6
a^8 b^4+14 a^6 b^6-11 a^4 b^8+3 a^2 b^10-a^10 c^2+13 a^8 b^2 c^2-14 a^6
b^4 c^2-6 a^4 b^6 c^2+7 a^2 b^8 c^2+b^10 c^2-6 a^8 c^4-14 a^6 b^2 c^4+34
a^4 b^4 c^4-10 a^2 b^6 c^4-4 b^8 c^4+14 a^6 c^6-6 a^4 b^2 c^6-10 a^2 b^4
c^6+6 b^6 c^6-11 a^4 c^8+7 a^2 b^2 c^8-4 b^4 c^8+3 a^2 c^10+b^2 c^10) y
z + .... = 0.

Angel Montesdeoca