HYACINTHOS 25200

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and HaHbHc, OaObOc the pedal triangles of H,O, resp.

Denote:

M1, M2, M3  = the midpoints of OHa, OHb, OHc, resp.

Ma, Mb, Mc = the midpoints of HOa, HOb, HOc, resp.

The Euler lines of M1MbMc, M2McMa, M3MaMb concur on the Euler line of ABC.

Denote

La, Lb, Lc = the Euler lines of M1MbMc, M2McMa, M3MaMb, resp.

PaPbPc = the pedal triangle of a point P on the Euler line.

1. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent.

2. The parallels to La, Lb, Lc through Pa, Pb, Pc, resp. are concurrent (?)

If true, which is the locus of the point of concurrence as P moves on the Euler line?

 

 

[César Lozada]:

 

 

> 1. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent.

At X(265) = REFLECTION OF X(3) IN X(125)

 

> 2. The parallels to La, Lb, Lc through Pa, Pb, Pc, resp. are concurrent (?)

Yes, on the polar trilinear of X(2407) through ETC’s: {30, 113, 1495, 1511, 1514, 1524, 1525, 1531, 1533, 1539, 1544, 1545, 1546, 1553, 1554, 1555, 1558, 1561, 1568, 2682, 2685, 2686, 3233, 3258, 5642, 6739, 10272, 10564, 11064}

 

If P is such that OP=t*OH and Q(P) is the point of concurrence of the indicated lines, then

X(113)Q(P) = -((9*R^2-2*SW) /(6*(3*R^2-SW)))*(t-3)*X(113) X(1495) and

 

Q(P) = (4*S^2*(9*R^2-2*SW)*t+3*(SB+ SC)*(S^2-3*SA^2))*(S^2-3*SB* SC) :: (barycentrics)

 

ETC pairs (P,Q(P)): (3,1511), (4,5642), (3146,113), (3627,10272), (4201,11064)

 

Others Q(P):

 

Q( G) = (4*S^2*(9*R^2-2*SW)+9*(SB+SC)* (S^2-3*SA^2))*(S^2-3*SB*SC) : :  (barycentrics)

= On lines: {30,113}, {110,3524}, {125,11539}, {541,10304}, {542,5054}, {5055,5972}, {6699,9143}

= midpoint of X(110) and X(3524)

= reflection of X(i) in X(j) for these (i,j): (125,11539), (5055,5972)

= [ 3.152797250531080, 1.80676613412734, 0.934689196497261 ]

 

Q(N) = (2*S^2*(9*R^2-2*SW)+3*(SB+SC)* (S^2-3*SA^2))*(S^2-3*SB*SC) : : (barycentrics)

= On lines: {30,113}, {110,549}, {125,10124}, {140,542}, {399,3524}, {547,5972}, {550,10706}, {2948,3653}, {3471,9214}, {3530,5609}, {3819,5663}, {5054,9143}, {5648,8263}, {5655,6030}, {9140,11539}

= midpoint of X(i) and X(j) for these {i,j}: {110,549}, {550,10706}, {1511,5642}, {5655,8703}, {9143,10264}

= reflection of X(i) in X(j) for these (i,j): (125,10124), (547,5972), (10272,5642)

= {X(5054), X(9143)}-Harmonic conjugate of X(10264)

= [ 2.806714408722111, 1.46159626759354, 1.333383492624800 ]

 

César Lozada