HYACINTHOS 28331

[Antreas P. Hatzipolakis]:

 

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

Denote:

Ab, Ac = the reflections of B', C' in PA', resp.
Bc, Ba = the reflections of C', A' in PB', resp.
Ca, Cb = the reflections of A', B' in PC', resp.

A*B*C* = the triangle bounded by AbAc, BcBa, CaCb

La =: PA*, Lb =: PB*, Lc =: PC*

1. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent.
Which is the point of concurrence in terms of P?

 

2. Which is the locus of P such the parallels to La, Lb, Lc through A', B', C', resp. are concurrent?

 

[César Lozada]:

 

 

1)

The entire plane. For P=u:v:w (trilinears) the point of concurrence is:

Q1(P) = a*(v*(a*w+c*u)*(S^2+SA*SC)-(S^2-SA*SC)*b*u*w)*(w*(a*v+b*u)*(S^2+SA*SB)-(S^2-SA*SB)*c*u*v) : :

= X(4)-vertex conjugate-of-X(P)

 

ETC pairs (P,Q1(P)): (1,3417), (2,3425), (3,4), (4,3), (5,15620), (6,3431), (15,16257), (16,16258), (20,5879), (24,254), (25,7612), (32,3406), (54,54), (58,947), (59,15381), (64,11270), (69,18532), (84,10623), (96,8884), (186,523), (249,2065), (250,10419), (251,5481), (252,11815), (254,24), (523,186), (947,58), (1138,3447), (1342,1343), (1343,1342), (2065,249), (3417,1), (3425,2), (3426,20421), (3431,6), (3432,3459), (3447,1138), (3459,3432), (3532,13452), (5481,251), (5879,20), (7612,25), (8883,14518), (8884,96), (10419,250), (10623,84), (11270,64), (11815,252), (13452,3532), (13472,14528) (incomplete list)

 

Q1( X(13) ) = ISOGONAL CONJUGATE OF X(20428)

= (SB+SC)*(SB*S^2+SA*SC*(SW+2*sqrt(3)*S))*(SC*S^2+SB*SA*(SW+2*sqrt(3)*S)) : : (barys)

= on the line {184, 9736}

= anticomplement of the complementary conjugate of X(13350)

= isogonal conjugate of X(20428)

= [ 6.5882290644921060, 3.5259174069973600, -1.8410763680870750 ]

 

Q1( X(14) ) = ISOGONAL CONJUGATE OF X(20429)

= (SB+SC)*(SB*S^2+SA*SC*(SW-2*sqrt(3)*S))*(SC*S^2+SB*SA*(SW-2*sqrt(3)*S)) : : (barys)

= on the line {184, 9735}

= anticomplement of the complementary conjugate of X(13349)

= isogonal conjugate of X(20429)

= [ -3.3030503972708560, -6.8715876096744820, 9.9224022411917260 ]

 

2)

Locus = {Linf } ∪ {Lemoine cubic K009} ∪ { circumquartic q4 through ETC’s 186}

q4: ∑[ y*z*(b^2*c^2*(-a^4*(-a^2+b^2+c^2)-(b^4+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*x^2+((-a^2+b^2+c^2)^2-b^2*c^2)*a^6*y*z)]=0, (barys).

 

ETC pairs (P, Q2(P)): (3,3), (4,52), (186,523)

 

Q2( X(32) ) = X(5)X(5007) ∩ X(6179)X(9764)

= a^6+2*(b^2+c^2)*a^4-(b^2+c^2)*b^2*c^2)*(2*a^6-6*c^2*a^2*b^2-(b^2+c^2)*a^4+(b^4-c^4)*(b^2-c^2) : : (barys)

= on lines: {5, 5007}, {6179, 9764}

= [ 0.6078614410695414, 0.0907698908144507, 3.2972723539268940 ]

 

César Lozada