HYACINTHOS 25502

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

A", B", C" = the reflections of I in BC, CA, AB, resp.

A'''B'''C''' = the orthic triangle of A"B"C"

Ma, Mb, Mc = the midpoints of the altitudes A"A''', B"B''', C"C''', resp.

R1 = the radical axis of the circles (Mb, MbB'), (Mc, McC')

R2 = the radical axis of the circles (Mc, McC'), (Ma, MaA')

R3 = the radical axis of the circles (Ma, MaA'), (Mb, MaB')

[the radical center of the circles is the point they concur at = the NPC center of A"B"C"]

Ra, Rb, Rc = the reflections of R1, R2, R3 in IA, IB, IC, resp.

A*B*C* = the triangle bounded by Ra,Rb,Rc

ABC, A*B*C* are parallelogic.

The parallelogic center (ABC, A*B*C*) lies on the OI line of ABC [ = Euler line of A"B"C"]

[Angel Montesdeoca]:

The parallelogic center (ABC, A*B*C*) is X(56)

 The parallelogic center (A*B*C*, ABC) is  (r-2R) X(1) + r X(4)
 
 V= ( a^3 (b+c)+a^2 (b^2-6 b c+c^2)-a (b-c)^2 (b+c)-(b^2-c^2)^2: ... : ... ).
 
   with (6-9-13)-search numbers (2.777677944199995442, 2.90381002649650, 0.348329258548341).
   V is the  midpoint of X(i) and X(j) for these {i,j}: {1,1479},{1837,2098}.  
   V is the reflection of X(i) in X(j) for these {i,j}: {10,3825}, {1210,496}, {4848,1210}, {5687,6700}, {6736,1329}-
  
  V lies on lines:  {1, 4}, {2, 1697}, {3, 10624}, {5, 7743}, {7, 738}, {8, 3452}, {10, 11}, {12, 3817}, {20, 1420}, {21, 3254}, {30, 4311}, {35, 6940}, {40, 3086}, {46, 10072}, {55, 474}, {56, 516}, {57, 962}, {63, 10529}, {65, 4301}, {78, 5853}, {102, 1067}, {142, 390}, {145, 908}, {165, 7288}, {329, 6762}, {354, 3671}, {355, 9669}, {411, 2078}, {495, 9955}, {496, 517}, {498, 6983}, {499, 5119}, {518, 10392}, {519, 1837}, {527, 11240}, {550, 5126}, {551, 2646}, {553, 3333}, {595, 1936}, {758, 10959}, {936, 5082}, {938, 3340}, {960, 3813}, {993, 10966}, {999, 4292}, {1000, 5818}, {1071, 1537}, {1108, 8804}, {1155, 5493}, {1193, 3755}, {1201, 3914}, {1319, 4297}, {1329, 3880}, {1385, 1387}, {1388, 9670}, {1482, 5722}, {1616, 3772}, {1698, 9819}, {1737, 5697}, {1770, 5563}, {1776, 6763}, {1788, 7991}, {1836, 3304}, {1858, 3874}, {1864, 3555}, {1898, 2801}, {2066, 8983}, {2099, 6738}, {2136, 7080}, {2269, 5257}, {2321, 3702}, {2478, 3872}, {2550, 8583}, {2551, 4853}, {3023, 11599}, {3085, 6964}, {3091, 9578}, {3146, 4308}, {3243, 5809}, {3244, 5048}, {3295, 5886}, {3303, 11375}, {3306, 10586}, {3338, 4031}, {3361, 3474}, {3478, 10570}, {3501, 8568}, {3576, 4294}, {3577, 5804}, {3582, 11010}, {3600, 9579}, {3612, 4309}, {3622, 4313}, {3624, 5218}, {3649, 4890}, {3660, 9943}, {3663, 3665}, {3687, 4673}, {3741, 10480}, {3746, 6946}, {3753, 9843}, {3814, 10915}, {3816, 5836}, {3847, 5123}, {3877, 5837}, {3878, 10916}, {3885, 4193}, {3889, 10394}, {3895, 5552}, {3913, 6745}, {3953, 7004}, {4035, 4742}, {4310, 4907}, {4315, 7354}, {4425, 8240}, {4654, 11037}, {4668, 8275}, {4863, 6743}, {5045, 10391}, {5049, 6147}, {5068, 7320}, {5084, 9623}, {5086, 10707}, {5128, 5435}, {5250, 5745}, {5252, 10863}, {5261, 9779}, {5265, 9778}, {5281, 5550}, {5289, 6737}, {5433, 10164}, {5533, 10265}, {5536, 7098}, {5570, 5884}, {5587, 10591}, {5687, 6700}, {5703, 10389}, {5758, 10396}, {5768, 7971}, {5794, 11235}, {6705, 10785}, {6767, 11374}, {6796, 11508}, {6975, 7741}, {7988, 10588}, {8715, 11502}, {8808, 10373}, {9956, 10593}, {10043, 10051}, {10543, 11263}, {10580, 11518}
 
 

VARIATION:

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

A", B", C" = the reflections of I in BC, CA, AB, resp.

A'''B'''C''' = the orthic triangle of A"B"C"

The circles having as diameters segments   A'A''', B'B''', C'C'''  are concurrent at X(11570) , studied in Hyacinthos 25048

Angel Montesdeoca