HYACINTHOS 24872

[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 AA',BB',CC', resp.

 

A",B",C"= the reflections of Ma,Mb,Mc in PA', PB', PC', resp.

ABC, A"B"C" are orthologic.

If P lies on the Euler line the orthologic center (MaMbMc, ABC) llies on the Euler line.

Which is the locus of the other orthologic center (ABC, MaMbMc) as P moves on the Euler line?

 

[César Lozada]:

Asked locus = Jerabek hyperbola

 

ETC-pairs (P, Za(P)=orthologic center (ABC,MaMbMc) ):

(2,3), (4,64), (5,54), (376,3532), (381,4), (403,1177), (3091,3527), (3545,6), (3839,3426), (5055,3431), (5066,1173)

 

if P is such that OP = t*OH then trilinear coordinates of P are:

Za(P) = a /(S^2-5*SB*SC-3*(S^2-3*SB*SC)* t) :  :

 

In general, Za(P) = isogonal-conjugate-of-Q, where Q is such that OQ=(-3*t+2)*OH.

 

Za(O) = a/(3*a^4-(b^2+c^2)*a^2-2*(b^2- c^2)^2) : : (trilinears)

= isogonal conjugate of X(382)

= On Jerabek hyperbola, cubic K850 and these lines: {2,3521}, {3,9544}, {6,3520}, {20,265}, {24,3426}, {54,1204}, {64,186}, {68,376}, {69,3528}, {73,5010}, {74,6759}, {185,3431}, {248,5206}, {378,3527}, {631,4846}, {1593,3531}, {3518,10606}, {3519,3522}, {5504,7689}, {6415,6449}, {6416,6450}

= [ 9.049236858447419, 8.65700987387075, -6.529220903901871 ]

 

For the  reciprocal orthologic center Zm(P):  if P is such that OP = t*OH then OZm(P)= ((3*t-1)/2)*OH.

ETC-pairs (P,Zm(P)):

(2,3), (3,550), (4,4), (5,140), (20,1657), (186,10295), (376,20), (381,5), (403,468), (428,6756), (546,3850), (547,3530), (549,548), (631,3522), (1598,7715), (3089,3517), (3090,3523), (3091,1656), (3146,5073), (3153,7574), (3524,376), (3529,5059), (3541,3516), (3542,3515), (3543,382), (3544,3533), (3545,2), (3830,3627), (3832,3851), (3839,381), (3843,3858), (3845,546), (3855,5056), (5054,8703), (5055,549), (5064,1595), (5066,3628), (5067,10299), (5071,631), (7552,7488), (7565,5576), (7576,3575), (7714,7487), (10201,1658), (10304,3534), (11001,3529)

 

Note: ABC and A”B”C” are orthologic for any P on the plane of ABC

 

Regards

César Lozada