HYACINTHOS 28573

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:
PaPbPc = the pedal triangle of P.

P1P2P3 = the reflection of PaPbPc in P
(ie P1, P2, P3 = the reflections of Pa, Pb, Pc in P, resp.)

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

M1, M2, M3 = the midpoints of MaP1, MbP2, McP3, resp.

Which is the locus of P such that ABC, M1M2M3 are perspective?
O,I lie on the locus.

 

[César Lozada]: 

 

The asked locus is the uncatalogued excentral-circum-cubic pK(X(6), X(140)) with barycentrics equation:

∑ [(3*S^2-SB*SC)*x*(c^2*y^2-b^2*z^2)] = 0

 

This cubic passes through ETC’s 1, 3, 4, 140, 1173, 3337, 7161, 15047.

 

ETC pairs (P, Q(P)=perspector (ABC, M1M2M3) ) = (1,5559), (3,550), (4,4), (3337,5557), (15047,140)

 

Some others:

 

Q( X(140) ) = X(4)X(11017) ∩ X(6)X(15720)

= (4*cos(C)^2-9)*(4*cos(B)^2-9)*cos(A) : : (trilinears)

= (-a^2+b^2+c^2)*((a^2-b^2+c^2)^2-9*a^2*c^2)*((a^2+b^2-c^2)^2-9*a^2*b^2) : : (barys)

= on the Jerabek hyperbola and  lines: {4, 11017}, {6, 15720}, {54, 15712}, {65, 5557}, {140, 1173}, {265, 5447}, {550, 16835}, {1216, 14861}, {1657, 22334}, {2889, 11592}, {3521, 3917}, {3522, 13452}, {3523, 13472}, {5562, 13623}, {7386, 14843}, {15321, 18553}, {15740, 23039}, {18296, 18531}

= [ 5.8869491127852830, 4.4699250147787540, -2.1709524265325820 ]

 

Q(X(1173)) = X(140)X(1173) ∩ X(3850)X(11703)

= (3*S^2-SA*SC)*(9*S^2+5*SB^2)*(3*S^2-SA*SB)*(9*S^2+5*SC^2) : : (barys)

= on lines: {140, 1173}, {3850, 11703}

= [ 0.0245506114532818, 0.0282097081033611, 3.6098036325497680 ]

 

Note: The isogonal conjugate of Q(X(140)) is:

Q^-1(X(140)) = EULER LINE INTERCEPT OF X(113)X(25714)

= (4*cos(A)^2-9)*cos(B)*cos(C) : : (trilinears)

= a^2*((-a^2+b^2+c^2)^2-9*b^2*c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : : (barys)

= 2*(4*R^2-SW)*X(3)+9*R^2*X(4)

= As a point on the Euler line, this center has Shinagawa coefficients (-4*F, 9*E+4*F)

= on lines: {2, 3}, {113, 25714}, {389, 12112}, {1173, 1199}, {1493, 10540}, {2914, 5609}, {3060, 15083}, {5007, 8744}, {5446, 15801}, {5943, 8718}, {6152, 16982}, {6243, 15052}, {6759, 11423}, {9781, 15032}, {12254, 16657}, {13353, 23060}, {13452, 22334}, {13474, 16835}, {14094, 16625}, {14853, 15581}, {15873, 16659}, {18296, 18532}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3091, 3146, 18531), (3091, 3547, 3090), (3518, 14865, 186)

= [ -2.7202167385540680, -3.5825606610108120, 7.3763834342475900 ]

 

César Lozada