HYACINTHOS 29470

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the pedal triangle of H (orthic triangle).

Denote:

La = the reflection of HA' in B'C'
Lb = the reflection of HB' in C'A'
Lc = the reflection of HC' in A'B'

Ma = the reflection of La in OA.
Mb = the reflection of Lb in OB.
Mc = the reflection of Lb in OC.

The lines Ma,Mb,Mc concur at a point Q. 

 

Denote:

A" = BC /\ Ma
B" = CA /\ Mb
C" = AB /\ Mc

The triangles ABC, A"B"C" are orthologic, with orthologic centers Q,S.

[César Lozada]:

Q = -4*b^4*a^6-2*c^2*a^6*b^2-4*c^4*a^6-2*a^10+2*b^6*c^4-3*c^2*b^8+c^10-3*c^8*b^2-2*c^8*a^2+2*c^6*a^4+5*a^8*c^2+b^10+2*c^6*b^4-2*c^4*b^2*a^4+2*c^6*b^2*a^2+2*c^2*b^6*a^2-2*c^2*b^4*a^4-2*b^8*a^2+2*b^6*a^4+5*a^8*b^2 : :
(Barycentrics)  

More simple form (barycentrics)
-2*a^10+(5*(c^2+b^2))*a^8-(c^2*(b^2+4*c^2)+b^2*(4*b^2+c^2))*a^6+(2*(c^2+b^2))*(b^2-c^2)^2*a^4+(b^2-c^2)^2*((c^2+b^2)^2-b^2*c^2)*a^2+(c^2+b^2)*(b^2-c^2)^4 : :  

[APH]:

Questions:

1. Properties of Q? (in order to be included in ETC, in the case it is not listed)

2. Coordinates/properties of S?

[César Lozada]:

 

Q = ANTICOMPLEMENT OF X(12134)

= 2*a^10-5*(b^2+c^2)*a^8+2*(2*b^4+b^2*c^2+2*c^4)*a^6-2*(b^4-c^4)*(b^2-c^2)*a^4+2*(b^6-c^6)*(b^2-c^2)*a^2-(b^4-c^4)*(b^2-c^2)^3 : : (barys)

= (SB+SC)*S^2+2*(R^2-SW)*SB*SC : : (barys)

= 2*X(4)-3*X(12022), 7*X(4)-12*X(12024), 3*X(4)-4*X(12241), 5*X(4)-4*X(16621), 7*X(4)-6*X(16654), 3*X(4)-2*X(16655), 9*X(4)-8*X(16656), 5*X(4)-6*X(16657), 4*X(4)-3*X(16658), 3*X(51)-2*X(13419), 4*X(143)-3*X(7540), 4*X(6146)-3*X(12022), 7*X(6146)-6*X(12024), 3*X(6146)-2*X(12241), 5*X(6146)-2*X(16621), 7*X(6146)-3*X(16654), 3*X(6146)-X(16655), 9*X(6146)-4*X(16656), 5*X(6146)-3*X(16657), 8*X(6146)-3*X(16658)

= lies on these lines: {2, 9707}, {3, 70}, {4, 6}, {5, 1614}, {11, 9638}, {20, 11411}, {22, 68}, {24, 1899}, {25, 18912}, {26, 3580}, {30, 5889}, {49, 13371}, {51, 13419}, {52, 10116}, {54, 427}, {74, 550}, {96, 98}, {110, 11585}, {125, 10018}, {140, 11464}, {143, 7540}, {154, 7505}, {156, 2072}, {182, 14788}, {184, 1594}, {185, 6152}, {235, 14157}, {265, 15761}, {343, 7512}, {378, 14216}, {382, 12174}, {389, 7576}, {403, 6759}, {428, 9781}, {468, 26882}, {539, 10625}, {542, 1205}, {546, 18394}, {548, 11468}, {568, 11819}, {569, 5133}, {578, 11550}, {858, 1147}, {1204, 10295}, {1209, 7495}, {1352, 7509}, {1370, 6193}, {1595, 15033}, {1656, 26864}, {1853, 19357}, {1885, 12290}, {1993, 14790}, {2888, 6636}, {2918, 2931}, {3043, 23315}, {3047, 23306}, {3060, 7553}, {3100, 12428}, {3147, 23291}, {3153, 22660}, {3515, 26944}, {3517, 26869}, {3518, 13567}, {3520, 6247}, {3541, 18925}, {3542, 11206}, {3564, 11412}, {3567, 11245}, {3575, 5890}, {3796, 7558}, {4296, 18970}, {5064, 11426}, {5422, 7528}, {5576, 14389}, {5663, 18563}, {5876, 13470}, {5944, 13561}, {6000, 18560}, {6143, 23332}, {6403, 26926}, {6515, 31305}, {6642, 18911}, {7383, 25406}, {7395, 18440}, {7403, 13434}, {7487, 18916}, {7502, 18356}, {7506, 18952}, {7507, 19347}, {7542, 23293}, {7583, 11462}, {7584, 11463}, {7667, 33523}, {7687, 14862}, {9544, 9820}, {9545, 31074}, {9730, 18128}, {9825, 15045}, {9862, 17401}, {9908, 26283}, {9920, 21284}, {10112, 29012}, {10114, 13417}, {10127, 15028}, {10192, 14940}, {10257, 11449}, {10263, 11264}, {10264, 15331}, {10574, 31833}, {10575, 17702}, {10594, 31383}, {10605, 17845}, {10619, 11430}, {11381, 13403}, {11413, 12118}, {11414, 12429}, {11441, 18531}, {11444, 31831}, {11455, 13488}, {11459, 12362}, {11461, 15171}, {11466, 11542}, {11467, 11543}, {11572, 18388}, {11645, 13598}, {11649, 11660}, {11750, 12225}, {12082, 32599}, {12111, 12605}, {12161, 31723}, {12278, 15072}, {13160, 18474}, {13171, 13564}, {13198, 32379}, {13367, 20299}, {13383, 26881}, {13491, 30522}, {13568, 18559}, {13619, 17846}, {15133, 20302}, {16238, 26913}, {16252, 16868}, {18390, 26883}, {18404, 32139}, {18445, 18569}, {18533, 18909}, {18583, 19123}, {18990, 19368}, {23335, 34148}, {26937, 32534}

= midpoint of X(6241) and X(12289)

= reflection of X(i) in X(j) for these (i,j): (4, 6146), (52, 10116), (382, 12370), (3575, 18914), (5876, 13470), (6152, 32377), (6240, 185), (6243, 32358), (6403, 26926), (7553, 13292), (10263, 11264), (11381, 13403), (12111, 12605), (12225, 11750), (12290, 1885), (13417, 10114), (14516, 3), (16654, 12024), (16655, 12241), (16658, 12022), (16659, 4), (18560, 21659)

= isogonal conjugate of S

= anticomplement of X(12134)

= X(6146)-of-anti-Euler triangle

= X(14516)-of-ABC-X3 reflections triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 1498, 32111), (4, 6146, 12022), (4, 6776, 7592), (4, 15032, 12233), (4, 16659, 16658), (1498, 18396, 4), (6146, 16655, 12241), (12022, 16659, 4), (12241, 16655, 4), (16621, 16657, 4)

= [ 11.6640123927716800, 14.3220366973378500, -11.6579820282979800 ]

 

S = ISOGONAL CONJUGATE OF Q

= a^2*(a^10-(2*b^2+3*c^2)*a^8+2*(b^4+b^2*c^2+c^4)*a^6-2*(2*b^6+b^4*c^2-c^6)*a^4+(b^4-c^4)*(5*b^4-2*b^2*c^2+3*c^4)*a^2-(2*b^4+b^2*c^2+c^4)*(b^2-c^2)^3)*(a^10-(3*b^2+2*c^2)*a^8+2*(b^4+b^2*c^2+c^4)*a^6+2*(b^6-b^2*c^4-2*c^6)*a^4-(b^4-c^4)*(3*b^4-2*b^2*c^2+5*c^4)*a^2+(b^4+b^2*c^2+2*c^4)*(b^2-c^2)^3) : : (barys)

= ((SA+SC)*S^2+2*(R^2-SW)*SA*SC)*((SA+SB)*S^2+2*(R^2-SW)*SA*SB) : : (barys)

= lies on these lines: {3, 8746}, {26, 394}, {52, 17974}, {97, 7512}, {1073, 7506}, {3518, 14919}, {3926, 31305}, {7528, 14376}

= isogonal conjugate of Q

= [ 146.5432603330903000, 119.3463220856046000, -146.6190632695528000 ]

 

César Lozada