HYACINTHOS 28330

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle, HaHbHc the pedal triangle of H, P a point and A'B'C' the pedal triangle of P.

Denote:

A2, A3 = the reflections of A' in PB', PC', resp.

B3, B1 = the reflections of B' in PC', PA', resp.

C1, C2 = the reflections of C' in PA', PB', resp.

A*B*C* = the triangle bounded by A2A3, B3B1, C1C2, resp.,

HaHbHc, A*B*C* are homothetic.
Homothetic center in terms of P ?


[César Lozada]:

For P=u:v:w (trilinears), the homothetic center Q(P) is

Q(P) = (2*cos(A)*u-cos(B)*v-cos(C)*w)*cos(B)*cos(C)  : :

 

Q( P* ∈ HP) = HP ∩ {orthic axis X(230)X(231)}. Therefore, P->Q(P)  maps all the points on the line HP on a fixed point on the orthic axis.

 

The appearance of (i, j) in the following list means Q(P ∈ HX(i) ) = X( j ):

(2, 468), (6, 1990), (9, 8756), (32, 6103), (39, 232), (96, 231), (230, 230), (523, 523), (566, 11062), (647, 647), (650, 650), (676, 676), (690, 16230), (1499, 2501), (1637, 1637), (1886, 1886), (2457, 7649), (2485, 2485), (2489, 2489), (2490, 2490), (2491, 2491), (2492, 2492), (2493, 2493), (2793, 14273), (2977, 2977), (3003, 3003), (3011, 3011), (3012, 3012), (3018, 3018), (3064, 3064), (3172, 16318), (3290, 3290), (3291, 3291), (3310, 3310), (3806, 3806), (4277, 14571), (4874, 4874), (5089, 5089), (6129, 6129), (6130, 6130), (6131, 6131), (6132, 6132), (6133, 6133), (6134, 6134), (6586, 6586), (6587, 6587), (6588, 6588), (6589, 6589), (6590, 6590), (6591, 6591), (6753, 6753), (7662, 7662), (8105, 8105), (8106, 8106), (8607, 8607), (8608, 8608), (8609, 8609), (8610, 8610), (8755, 8755), (8758, 8758), (9125, 9125), (9189, 9189), (9209, 9209), (9465, 14580), (10418, 10418), (11176, 11176), (11657, 11657), (12077, 12077), (13186, 13186), (13400, 13400), (13531, 16328), (14276, 14276), (14325, 14325), (14326, 14326), (14425, 14425), (16040, 16040), (16272, 16272), (16303, 16303), (16304, 16304), (16305, 16305), (16306, 16306), (16307, 16307), (16308, 16308), (16309, 16309), (16310, 16310), (16311, 16311), (16312, 16312), (16313, 16313), (16314, 16314), (16315, 16315), (16316, 16316), (16317, 16317), (16319, 16319), (16320, 16320), (16321, 16321), (16322, 16322), (16323, 16323), (16324, 16324), (16325, 16325), (16326, 16326), (16327, 16327), (16329, 16329), (16330, 16330), (16331, 16331), (16332, 16332), (16333, 16333), (16334, 16334), (16335, 16335), (21348, 21348)

 

Some others:

Q( HX(1) ) = ORTHIC AXIS INTERCEPT OF X(1)X(4)

= (a^2-b^2+c^2)*(a^2+b^2-c^2)*(2*a^2-(b+c)*a-(b-c)^2) : : (barys)

= on lines: {1, 4}, {25, 18613}, {108, 2078}, {230, 231}, {354, 1824}, {415, 648}, {429, 15888}, {517, 1835}, {519, 860}, {551, 5136}, {942, 1825}, {971, 6357}, {1060, 1074}, {1062, 1076}, {1420, 14257}, {1426, 3057}, {1430, 8750}, {1465, 15252}, {1758, 9357}, {1826, 16777}, {1830, 1876}, {1842, 11363}, {1861, 1897}, {1865, 3723}, {1874, 17724}, {1880, 17602}, {1936, 7012}, {3304, 4185}, {3746, 7414}, {4200, 11240}, {4654, 5733}, {4658, 14016}, {5231, 7046}, {6851, 9643}, {8606, 10902}, {10295, 11809}, {11518, 14018}

= polar conjugate of X(1121)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1068, 225), (1785, 1870, 1877), (1897, 17923, 1861), (3011, 3012, 8758), (8758, 16272, 3012)

= [ 0.8239835546647523, 0.7234920685129390, 2.7594852553993810 ]

 

Q( HX(11) ) = ORTHIC AXIS INTERCEPT OF X(4)X(11)

= (a^2+b^2-c^2)*(2*a^5-2*(b+c)*a^4-(3*b^2-8*b*c+3*c^2)*a^3+3*(b^2-c^2)*(b-c)*a^2+(b-c)^4*a-(b^4-c^4)*(b-c))*(a^2-b^2+c^2) : : (barys)

= on lines: {4, 11}, {33, 17728}, {105, 4232}, {140, 15252}, {208, 11376}, {230, 231}, {318, 6691}, {406, 3304}, {451, 15888}, {528, 4242}, {1387, 1845}, {1785, 15325}, {1897, 3035}, {2968, 6717}, {3515, 14667}, {5094, 20621}, {5433, 7952}, {6713, 21664}, {7079, 13609}, {15253, 21841}

= [ 0.6039078407790179, 0.9452350226553834, 2.7075443089403360 ]

 

Q( HX(13) ) = ORTHIC AXIS INTERCEPT OF X(4)X(13)

= SB*SC*(3*S^2+sqrt(3)*(3*SA-SW)*S-9*SB*SC) : : (barys)

= on lines: {4, 13}, {230, 231}, {462, 8754}, {471, 648}, {6111, 10295}, {8738, 17983}

= polar conjugate of the isotomic conjugate of X(530)

= [ 0.7257399391206414, 0.8224799538717777, 2.7362983880943910 ]

 

Q( HX(14) ) = ORTHIC AXIS INTERCEPT OF X(4)X(14)

= SB*SC*(3*S^2-sqrt(3)*(3*SA-SW)*S-9*SB*SC) : : (barys)

= on lines: {4, 14}, {230, 231}, {463, 8754}, {470, 648}, {6110, 10295}, {8737, 17983}

= polar conjugate of the isotomic conjugate of X(531)

= [ 0.2632385906920427, 1.2884851003945330, 2.6271416013149130 ]

 

Q( HX(15) ) = ORTHIC AXIS INTERCEPT OF X(4)X(15)

= SB*SC*(3*SA-sqrt(3)*S)*(3*SB+3*SC+2*sqrt(3)*S) : : (barys)

= on lines: {4, 15}, {186, 6104}, {230, 231}, {299, 340}, {396, 463}, {397, 13367}, {403, 6107}, {470, 8737}

= [ 0.8843733376841744, 0.6626447871373093, 2.7737380888043150 ]

 

Q( HX(16) ) = ORTHIC AXIS INTERCEPT OF X(4)X(16)

= SB*SC*(3*SA+sqrt(3)*S)*(3*SB+3*SC-2*sqrt(3)*S) : : (barys)

= on lines: {4, 16}, {186, 6105}, {230, 231}, {298, 340}, {395, 462}, {398, 13367}, {403, 6106}, {471, 8738}

= [ -21.1770877614444800, 22.8912381673199700, -2.4330752824935300 ]

 

César Lozada