HYACINTHOS 26883

[Antreas P. Hatzipolakis]:

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

Denote:

Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

M1, M2, M3 = the midpoints of MaNa, MbNb, McNc, resp.

Which is the locus of P such that A'B'C', M1M2M3 are orthologic?
The entire plane?

[Angel Montesdeoca]:

*** The locus of P such that A'B'C', M1M2M3 are orthologic is  the entire plane.

*** When P traverses the circumcircle the the locus of orthologic center of A'B'C' with respect to M1M2M3     is the circumcenter.

*** When P traverses the circumcircle the the locus of orthologic center of M1M2M3 with respect to  A'B'C'  [the centroid of {A,B,C,P}] is the circle of barycentric equation:

(a^2-3 b^2-3 c^2) x^2+(-2 a^2-2 b^2+10 c^2) x y+(-3 a^2+b^2-3 c^2) y^2+(-2 a^2+10 b^2-2 c^2) x z+(10 a^2-2 b^2-2 c^2) y z+(-3 a^2-3 b^2+c^2) z^2=0.

The center of this circle is X(140), midpoint of X(3) and X(5),   and passes through 620, 3035, 5972, 6036, 6699, 6710, 6711, 6712, 6713, 6714, 6715, 6716, 6717, 6718, 6719, 6720, 10120, 13372, 15240.

The perspector  of this circle is W = X(140)X(524) /\ X(468)X(3055)

W = 1/(a^4-3 a^2 (b^2+c^2)+2 b^4-5 b^2 c^2+2 c^4) :  ... : ....,

W lies on lines X(i)X(j) for these {i, j}: {140,524}, {468,3055}, {1232,3266}.
And with  (6 - 9 - 13) - search numbers:  (2.24135980865415, 2.55872522568660, 0.834765567822514).

 
Another description of this circle:
Let P be a point on the circumcircle. The bicevian conic of X(2) and P is a rectangular hyperbola, H. Let X be the center of H. As P varies, X traces a circle centered at X(140). (Randy Hutson, November 2, 2017) .

Angel Montesdeoca