HYACINTHOS 26214

[Antreas P. Hatzipolakis]:

 

 

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

Denote:

Ma, Mb, Mc = the midpoints of AP, BP, CP, resp.

A" = PG /\ B'C', B" = PG /\ C'A', C" = PG /\ A'B'

Which is the locus of P such that MaMbMc, A"B"C" are perspective?

The entire plane ?

I lies on the locus. The perspector is the Feuerbach point (*)

(*) Kadir Altintas Here and Here

Loci:

1. Which is the locus of the perspectors as P moves on the Euler line?

 

2. Which is the locus of P such that the perspectors lie on the NPC?
I,O,H lie on the locus.

[César Lozada]:

 

>Which is the locus of P such that MaMbMc, A"B"C" are perspective? The entire plane ?

Yes. For P=u:v:w (trilinears) the perspector Q(P) is:

Q(P) = (b*v+c*w-a*u)*(b*v-c*w)^2/a : :

 

ETC-pairs(P,Q(P)):

(1,11), (3,125), (4,122), (5,2972), (6,125), (8,3756), (9,11), (10,244), (11,3126), (30,1650), (37,244), (39,3124), (57,5514), (63,6506), (69,6388), (75,6377), (86,6627), (114,868), (115,1649), (120,3675), (141,3124), (142,3119), (216,2972), (223,5514), (230,868), (323,10413), (440,4466), (514,6544), (519,1647), (523,1649), (524,1648), (536,1646), (538,1645), (597,8288), (599,6791), (650,3126), (1086,6544), (1125,3120), (1210,7004), (1211,3125), (1212,3119), (1213,3120), (1249,122), (1645,888), (1646,891), (1647,900), (1648,690), (1649,690), (1650,9033), (2482,1648), (3117,6784), (3150,684), (3161,3756), (3163,1650), (3229,2086), (3290,3675), (3452,2170), (3580,2088), (3666,3125), (3720,2486), (3739,3121), (3741,3122), (3752,2170), (3772,7117), (3840,3123), (4370,1647), (5664,1637), (6337,6388), (6376,6377), (6505,6506), (6509,3269), (6544,900), (6626,6627), (11019,2310), (11165,6791), (13466,1646), (13567,3269)

 

Note that:
Q(P) = Q( Complement(Isotomic( Anticomplement(P))) )

 

Q(X(7)) = complement of X(658)

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

= On lines: {2,658}, {9,1768}, {11,1146}, {124,1566}, {1212,5316}, {1615,5658}, {2632,6587}, {2968,3239}, {5328,6554}

= {X(11), X(3119)}-Harmonic conjugate of X(1146)

= [ 3.897581749361377, 2.67468951050455, -0.009927525070174 ]

 

> 1. Which is the locus of the perspectors as P moves on the Euler line?

Locus = Line {122,125}

Q(X(20)) = (b^2-c^2)^2*(-a^2+b^2+c^2)^2*( 5*a^4-2*(b^2+c^2)*a^2-3*(b^2- c^2)^2) : : (barycentrics)

= On lines: {3,6723}, {4,3184}, {122,125}, {136,3154}

= {X(125), X(1650)}-Harmonic conjugate of X(122)

= [ 3.691425320251953, 2.90404414347690, -0.073562380615913 ]

 

> 2. Which is the locus of P such that the perspectors lie on the NPC?. I,O,H lie on the locus.

Locus = {medians-of-ABC} \/ {Thomson cubic K002}

ETC-pairs (P,Q(P)): (1,11), (3,125), (4,122), (6,125), (9,11), (57,5514), (223,5514), (1249,122)

 

Q( X(282)) = (-a+b+c)*(a^6-2*(b+c)*a^5-(b+ c)^2*a^4+4*(b^3+c^3)*a^3-(b^2- c^2)^2*a^2-2*(b^4-c^4)*(b-c)* a+(b^2-c^2)^2*(b+c)^2)*(b-c)^ 2*(a^3+(b+c)*a^2-(b+c)^2*a-(b^ 2-c^2)*(b-c))^2 : : (barycentrics)

= On the nine-points circle and these lines: {117,6260}, {124,7358}, {7952,10271}

= [ 3.161083596931126, 3.24784597939545, -0.066805933180682 ]

 

Q(X(1073)) = sin(2*A)*sin(B-C)^2*(3*cos(A)- cos(B-C))^2*((2*cos(2*A)+6)* cos(B-C)-9*cos(A)+cos(3*A)) : : (trilinears)

= On the nine-points circle and these lines: {122,1562}, {132,2883}, {133,1249}

= [ 2.381330158953369, 3.38967959766687, 0.194887764005760 ]

 

César Lozada