HYACINTHOS 28802

[Antreas P. Hatzipolakis]:

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

Denote:

(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.

(Nab), (Nac) = the reflections of (Nb), (Nc) in PNa, resp.
(Nbc), (Nba) = the reflections of (Nc), (Na) in PNb, resp.
(Nca), (Ncb) = the reflections of (Na), (Nb) in PNc, resp.

Ra, Rb, Rc = the radical axes of ((Nba), (Nca)), ((Ncb), (Nab)),
((Nac), (Nbc)), resp, [concurrent at P]
[Note: Since (Nba) = (Nca), (Ncb) = (Nab), (Nac) = (Nbc) we have that
Ra, Rb, Rc = the perpendicular bisectors of NbaNca, NcbNab, NacNbc,
resp.]

Which is the locus of P such that:

1. the reflections of Ra, Rb, Rc in BC, CA, AB, resp. are concurrent?
N lies on the locus (they are parallels)

2. the parallels to Ra, Rb, Rc through A, B, C, resp. are concurrent?
I lies on the locus.

3. the parallels to Ra, Rb, Rc through A', B, C', resp. are concurrent?
I lies on the locus.

 

 [César Lozada]:

 

Loci are very hard to be calculated.

 

Particular cases:

1)      Some points P in the locus with Q1(P) point of concurrence:

Q1(X(5)) = X(1154)

 

Q1(X(17)) = MIDPOINT OF X(16) AND X(5238)

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

= lies on these lines: {3, 6}, {17, 10616}, {30, 22891}, {533, 14144}, {617, 16530}, {2004, 3131}

= midpoint of X(16) and X(5238)

= reflection of X(17) in X(10616)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 19780, 3107), (16, 21159, 5351)

= [ -1.4812801132967960, -0.3934150369371020, 4.5966964828470490 ]

 

Q1(X(18)) = MIDPOINT OF X(15) AND X(5237)

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

= lies on these lines: {3, 6}, {18, 10617}, {30, 22846}, {532, 14145}, {616, 16529}, {2005, 3132}

= midpoint of X(15) and X(5237)

= reflection of X(18) in X(10617)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 19781, 3106), (15, 21158, 5352)

= [ -60.0005919594745200, -44.9247250774516700, 62.4349781322084100 ]

 

2)      Some points P in the locus with Q2(P) point of concurrence:

Q2(X(1)) = X(104)

Q2(X(3)) = X(3)

Q2(X(17)) = X(14)

Q2(X(18)) = X(13)

 

3)      Some points P in the locus with Q3(P) point of concurrence:

Q3(X(1)) = X(11570)

Q3(X(3)) = X(5)

 

César Lozada