HYACINTHOS 28465

 

[Antreas P. Hatzipolakis]:

 

 

 

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.

D = the Poncelet point of ABCN ( = X(137)

A'B'C' = the pedal triangle of D wrt triangle NaNbNc

Ma, Mb, Mc = the midpoints of A'Na, B'Nb, C'Nc, resp.

1. ABC, A'B'C' are homothetic. (Hyacinthos 28460)

2. ABC, MaMbMc are orthologic.

3. A'B'C', MaMbMc are orthologic

 

Orthologic centers?

 

 

[César Lozada]:

 

1)

X(25147)

 

2)

 

A->Ma = X(25043)

 

Ma->A = X(5)X(49) ∩ X(137)X(547)

= 20*S^4-(87*R^4+10*R^2*(SA-8*SW)-4*SA^2-12*SB*SC+20*SW^2)*S^2-(27*R^4-26*R^2*SW+8*SW^2)*SB*SC : : (barys)

= 7*X(5)+X(1141), 3*X(5)+X(12026), X(128)-5*X(12812), X(137)+3*X(547), X(140)+3*X(23516), X(930)-9*X(15699), 3*X(1141)-7*X(12026), X(1263)+7*X(3090), 5*X(1656)-X(6592), 15*X(1656)+X(11671), 5*X(1656)+3*X(25147), 3*X(3628)-X(13372), 3*X(6592)+X(11671), X(6592)+3*X(25147), X(11671)-9*X(25147)

= on lines: {5, 49}, {128, 12812}, {137, 547}, {140, 23516}, {930, 15699}, {1209, 20413}, {1263, 3090}, {1656, 6592}, {3628, 13372}, {5055, 14072}, {5056, 23237}, {7486, 13512}, {10096, 15366}, {14652, 21308}

= {X(1656), X(25147)}-harmonic conjugate of X(6592)

= [ 0.4119030560701827, -0.0865448695127290, 3.5104710579992610 ]

 

3)

 

A’ -> Ma = X(137)X(1209) ∩ X(547)X(18807)

= (S^2+SB*SC)*(8*S^4+(R^2*(41*R^2+2*SA-26*SW)+4*SA^2+4*SB*SC)*S^2-90*R^8+6*(11*SA+28*SW)*R^6+(14*SA^2-84*SA*SW-121*SW^2)*R^4-2*(4*SA^2-17*SA*SW-19*SW^2)*SW*R^2-4*(SA+SW)*SW^3) : : (barys)

= on lines: {137, 1209}, {547, 18807}, {3628, 25150}, {25043, 25147}

= [ -4.1468004846064100, -2.4733304796346590, 7.2668781145497950 ]

 

Ma->A’ = Ma->A

 

César Lozada