HYACINTHOS 25735

[Tran Quang Hung]:


Dear Mr César Lozada,

 

I see the similar problem with incenter I.

 

Let ABC be a triangle. Orthocenter H.

 

A'B'C' is cevian triangle of incenter I.

 

A'',B'',C'' are midpoints of AA',BB',CC'.

 

Then isogonal conjugate of I wrt triangle A''B''C'' lies on line IH.

 

Which is this point ?

 

General problem on McKay cubic

 

Let ABC be a triangle. Orthocenter H and P is a point on McKay cubic

 

A'B'C' is cevian triangle of a point P.

 

A'',B'',C'' are midpoints of AA',BB',CC'.

 

Then isogonal conjugate of P wrt triangle A''B''C'' lies on line PH.

 

[César Lozada]:

 

Dear Mr. Hung,

 

Indeed, the locus of P such that P”(P), the isogonal conjugate of P wrt A”B”C”, lies on the line HP is the McCay cubic.

 

P”(O) = See Hyacinthos #25728

 

P”(H) = X(185)

 

P”(I) = X(1)X(4) ∩ X(10)X(75)

= (b+c)*a^3+(b^2+c^2)*a^2+(b+c)* (b^2+c^2)*a+(b^2-c^2)^2 :: (barycentrics)

= SW*X(1)-2*r^2*X(4) = 2*(r^2-s^2)*X(10)+(2*s^2-SW)*X (75)

= On lines: {1,4}, {2,988}, {7,989}, {8,3914}, {9,5286}, {10,75}, {12,3666}, {19,1773}, {21,3011}, {30,5266}, {37,5254}, {38,6734}, {40,1423}, {63,5230}, {65,3782}, {141,3714}, {171,4292}, {238,12572}, {315,3879}, {377,612}, {387,3751}, {443,5268}, {495,3931}, {516,5255}, {518,1834}, {519,5015}, {527,1046}, {536,3704}, {611,5706}, {614,2478}, {908,1193}, {938,4310}, {940,10404}, {958,3772}, {960,4415}, {978,3452}, {982,1210}, {1074,3085}, {1076,4293}, {1086,3812}, {1100,7745}, {1329,3752}, {1330,5847}, {1469,10441}, {1722,2551}, {1737,3670}, {1770,5264}, {1836,5710}, {2475,3920}, {2782,5988}, {3120,10459}, {3146,4339}, {3339,4862}, {3361,7397}, {3664,4911}, {3672,5261}, {3674,4920}, {3677,9581}, {3701,4202}, {3744,6284}, {3749,4294}, {3891,5016}, {3912,6656}, {3947,7377}, {3976,11019}, {4104,9534}, {4187,5121}, {4201,7081}, {4298,6996}, {4352,7179}, {4416,7754}, {4642,6735}, {4646,12607}, {4696,4972}, {4850,11681}, {4851,7784}, {4902,5586}, {4968,5051}, {5080,5262}, {5084,5272}, {5247,12527}, {5269,9579}, {5393,7389}, {5405,7388}, {5725,9654}, {7383,10320}, {7395,8071}, {7399,10523}, {7613,11024}, {8069,11414}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1,3944,946), (10,3663,986), (12,3666,5530), (225,5307,1838), (341,4429,10), (2551,4000,1722)

= [ 1.996944427058847, 2.03251341176473, 1.311873153581637 ]

 

César Lozada