HYACINTHOS 24916

 [Antreas P. Hatzipolakis]:

 

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

Denote:

Ma, Mb, Mc = the midpoints of PA', PB', PC', resp.

M1, M2, M3 = the midpoints of AA', BB', CC', resp.

 For P = N:

The reflections of MaM1, MbM2, McM3 in NA', NB', NC', resp. concur on the Euler line of ABC. Point?

Locus?

Cf. Hyacinthos 23901

 

[César Lozada]:

Locus = Gibert´s K060, through ETC’s: 4, 5, 13, 14, 30, 79, 80, 265, 621, 622, 1117, 1141, 5627, 6761

 

ETC pairs (P, Z(P)=point of concurrence): (4,4), (5,3628), (30,30), (265,30)ç

 

Some others:

 

Z(X(13)) = Midpoint of X(13) and  X(396)

= 3*(2*a^6-5*(b^2+c^2)*a^4+2*(b^ 4-4*b^2*c^2+c^4)*a^2+(b^4-c^4) *(b^2-c^2))-2*sqrt(3)*(2*a^4+ 5*(b^2+c^2)*a^2-3*(b^2-c^2)^2) *S : : (barycentrics)

= sqrt(3)*S*X(5)+SW*X(6)

= midpoint of X(i) and X(j) for these {i,j}: {13,396}, {15,5318}, {115,6783}, {5472,6115}

= on cubic K369 and these lines: {3,5335}, {5,6}, {11,5357}, {12,5353}, {13,15}, {14,5066}, {16,17}, {61,546}, {62,3628}, {115,6783}, {230,5472}, {303,6390}, {381,5334}, {395,547}, {398,3850}, {442,5362}, {468,10633}, {524,623}, {530,6671}, {532,6669}, {548,10645}, {550,5340}, {624,3589}, {635,3631}, {1503,7684}, {1657,5344}, {1990,6117}, {2041,6221}, {2042,6398}, {2045,8976}, {3412,3853}, {3530,10646}, {3575,10632}, {3580,8838}, {3858,5339}, {4187,5367}, {5073,5366}, {5617,9112}, {6107,6113}, {6114,9300}, {6116,6748}, {6676,10635}, {6756,10641}

= [ 0.832569869017162, 0.83076511382802, 2.681256386633980 ]

 

Z(X(14)) = Midpoint of X(14) and  X(395)

= 3*(2*a^6-5*(b^2+c^2)*a^4+2*(b^ 4-4*b^2*c^2+c^4)*a^2+(b^4-c^4) *(b^2-c^2))+2*sqrt(3)*(2*a^4+ 5*(b^2+c^2)*a^2-3*(b^2-c^2)^2) *S : : (barycentrics)

= -sqrt(3)*S*X(5)+SW*X(6)

= midpoint of X(i) and X(j) for these {i,j}: {14,395}, {16,5321}, {115,6782}, {5471,6114}

= on cubic K369 and these lines: {3,5334}, {5,6}, {11,5353}, {12,5357}, {13,5066}, {14,16}, {15,18}, {61,3628}, {62,546}, {115,6782}, {230,5471}, {302,6390}, {381,5335}, {396,547}, {397,3850}, {442,5367}, {468,10632}, {524,624}, {531,6672}, {533,6670}, {548,10646}, {550,5339}, {623,3589}, {636,3631}, {1503,7685}, {1657,5343}, {1990,6116}, {2041,6398}, {2042,6221}, {2046,8976}, {3411,3853}, {3530,10645}, {3575,10633}, {3580,8836}, {3858,5340}, {4187,5362}, {5073,5365}, {5613,9113}, {6106,6112}, {6115,9300}, {6117,6748}, {6676,10634}, {6756,10642

= [ 1.584020955050334, 3.91738195466788, 0.197544226344922 ]

 

Z(X(79)) = 2*a^4+4*(b+c)*a^3+(b^2+4*b*c+ c^2)*a^2-4*(b^2-c^2)*(b-c)*a- 3*(b^2-c^2)^2 : : (barycentrics)

= complement of X(3650)

= midpoint of X(79) and X(3649),…

= On lines: {1,30}, {2,3650}, {5,5221}, {7,496}, {57,3652}, {142,3647}, {191,7308}, {226,3579}, {329,442}, {388,8148}, {495,4295}, {546,5902}, {553,9955}, {758,3626}, {954,3651}, {1159,5229}, {1387,4298}, {1483,9657}, {1770,5719},…

= [ 0.457762856848009, 0.46101434340096, 3.110225541007708 ]

 

Z(X(80)) = 2*a^4-4*(b+c)*a^3+(b^2+4*b*c+ c^2)*a^2+4*(b^2-c^2)*(b-c)*a- 3*(b^2-c^2)^2 : : (barycentrics)

= On lines: {1,3628}, {5,2099}, {8,496}, {10,6675}, {11,5844}, {12,5425}, {30,80}, {57,355}, {140,10950}, {495,3475}, {499,1483}, {515,5122}, {519,1387}, {758,6797}, {952,1319}, {999,6946}, {1058,4678}, {1159,10590}, {1317,3582}, {1482,10593}, {1837,5119},…

= [ 0.986750779080938, 2.45286479164664, 1.487103881960890 ]

 

Regards,

César Lozada