HYACINTHOS 26413

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C', A"B"C" the cevian, circumcevian triangles of H, resp.

Let A* be the second intersection (other than A") of the circumcircles of A"B'C' and ABC

Let B* be the second intersection (other than B") of the circumcircles of A'B"C' and ABC

Let C* be the second intersection (other than C") of the circumcircles of A'B'C" and ABC.

1. ABC, A*B*C* are perspective

2. the parallels to AA*,BB*,CC* through A', B', C', resp. are concurrent

 

[César Lozada]:

 

1)      X(6)

2)      X(1843)

César Lozada

 

[Antreas P. Hatzipolakis]:

 

That is:

Let A'B'C' be the cevian triangle of H and K the symmedian point X(6).

The parallels to AK, BK, CK through A',B',C', resp. are concurrent (at X(1843)).

Locus:

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

Which is the locus of P such that the parallels to AP, BP, CP through A', B', C', resp. are concurrent?

The Jerabek hyperbola + ?

[César Lozada]:

 

Locus = {Jh =Jerabek hyperbola} \/ {Line-at-infinity}

 

ETC pairs (P,Q(P)=point of concurrence) for P on Jh:

(3,52), (4,4), (6,1843), (54,6152), (64,185), (65,1829), (66,6), (67,5095), (68,155), (69,193), (70,2904), (74,1986), (265,113), (1173,11817), (2574,2574), (2575,2575), (3519,13431), (6145,3574), (11564,10294), (11744,13202)

 

Q(P on line-at-infinity)=P

 

For P on Jh:

 

Q(P) lies on the line {P, 6}

 

Locus of Q(P) = {  A rectangular hyperbola Qh : (SB-SC)*(SA^2*x^2+SB*SC*y*z)+… = 0}

Qh passes through the vertices of the orthic and the cevian-of-X(648) triangles and ETC’s: 4, 6, 52, 113, 155, 185, 193, 1162, 1163, 1829, 1839, 1843, 1858, 1986, 2574, 2575, 2904, 2905, 2906, 2907, 2914, 3574, 5095, 5895, 6152, 10294, 11817, 13202, 13420, 13431

 

Qhc = Center of Qh = X(1112)

 

Qhp = Perspector of Qh =

= SB*SC /(5*S^2+SA^2+2*SB*SC-2*SW^2) : : (barycentrics)

= isogonal conjugate of {2,1632}/\ {3,895}

= [ -2.299857178153620, -1.80848905563851, 5.954167910343480 ]

 

Some Q(P):

 

Q(X(71)) = X(4)X(916) ∩ X(6)X(31)

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

= On lines: {4,916}, {6,31}, {185,516}, {1843,9028}, {2772,13202} 

= [ -1.704943548378464, -0.12253763679448, 4.512395252786005 ]

 

Q(X(72)) = X(1)X(6) ∩ X(4)X(912)

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

= On lines: {1,6}, {4,912}, {46,2900}, {65,3419}, {79,6598}, {145,6987}, {185,517}, {210,10198}, {226,3874}, {442,942}, {758,950}, {916,1902}, {1490,12704}, {1708,3811}, {2000,5707}, {2771,12690}, {2906,3193}, {2949,10902}, {3218,3651}, {3487,3873}, {3488,3869}, {3574,5777}, {3586,3901}, {3870,10267}, {3894,9612}, {3916,10391}, {3927,13615}, {4018,5895}, {4430,6846}, {4661,10587}, {5082,7672}, {5439,5705}, {5715,5927}, {5745,10122}, {5758,12116}, {6857,11020}, {6889,10202}, {7483,11018}, {11012,12675}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (9, 5904, 72), (72, 5728, 405), (1708, 3811, 11517), (5904, 10399, 9)

= [ -3.194067195429782, 0.28294601821846, 4.918963636416115 ]

 

Q(X(73)) = X(4)X(5906) ∩ X(6)X(41)

= a^2*((b^2+c^2)*a^4+(b^3+c^3)* a^3-(b^3-c^3)*(b-c)*a^2-(b^3- c^3)*(b^2-c^2)*a-(b^2-c^2)^2* b*c)*((b+c)*a^4-b*c*a^3-(b+c)* (2*b^2-b*c+2*c^2)*a^2+b*c*(b+ c)^2*a+(b^3-c^3)*(b^2-c^2)) : : (barys)

= On lines: {4,5906}, {6,41}, {185,515}

= [ -3.929608063582715, -4.86562372074262, 8.822838471767447 ]

 

César Lozada