HYACINTHOS 25463

[César Lozada]:

 

If P moves on the circumcircle of ABC then the centroid G(P) of the pedal triangle of P moves on an ellipse {E} with center G=X(2) . This ellipse passes through  the vertices of the pedal triangle of X(376) = centroid of the antipedal triangle of G, and through ETC centers X(125), X(3756), X(5642), X(6784), X(6786), X(6791), X(6793).

 

Let’s say that P (on the circumcircle) is the anti-centroidal-pedal-of-G(P)

 

Conjecture: {E} passes through the vertices of the orthic triangle of Thomson triangle.

 

Correspondent centers:

X(125) = centroid of the pedal triangle of X(74)

X(3756) = centroid of the pedal triangle of X(106)

X(5642) = centroid of the pedal triangle of X(110)

X(6784) = centroid of the pedal triangle of X(98)

X(6786) = centroid of the pedal triangle of X(99)

X(6791) = centroid of the pedal triangle of X(111)

X(6793) = centroid of the pedal triangle of X(112)

 

More centers on {E}:  (Antipodes of known centers on it)

 

Reflection of X(125) in X(2) = X(5642)

Reflection of X(6784) in X(2) = X(6786)

 

Reflection of X(3756) in X(2)

= (2*a-b-c)*(2*a^2-3*(b+c)*a -b^2+6*b*c-c^2) : : (barycentrics)

= 2*X(3699)+X(3756)

= on lines: {2,1280}, {121,519}, {524,5205}, {900,1635}, {952,10713}, {1086,9458}, {1213,6791}, {1647,4152}, {3679,5854}

= midpoint of X(2) and X(3699)

= reflection of X(3756) in X(2)

= Tripolar centroid of X(2415)

= anti-centroidal-pedal of X(1293)

= [ 6.186750099973455, -0.28355937958212, 0.981551698553481 ]

 

Reflection of X(6791) in X(2)

= (2*a^2-b^2-c^2)*(2*a^4-5*(b^2+ c^2)*a^2-b^4+10*b^2*c^2-c^4) : :  (barycentrics)

= on lines: {2,5503}, {125,599}, {126,524}, {351,690}, {538,9127}, {542,10717}, {543,5108}, {1992,4563}, {5477,8030}, {5650,6784}, {5969,9172}, {6786,9023}

= midpoint of X(2) and X(9146)

= reflection of X(6791) in X(2)

= Tripolar centroid of X(2418)

= anti-centroidal-pedal of X(1296)

= [ 6.215350836466692, -0.65629905924602, 1.226401906093147 ]

 

(b^2-c^2)^2*(-a^2+b^2+c^2)*(a^ 4+(2*(b^2+c^2))*a^2-2*b^2*c^2- 3*c^4-3*b^4)

 

Reflection of X(6793) in X(2)

= (b^2-c^2)^2*(-a^2+b^2+c^2)*(a^ 4+(2*(b^2+c^2))*a^2-2*b^2*c^2- 3*c^4-3*b^4): :  (barycentrics)

= 4*X(127)-X(1562)

= on lines: {2,6793}, {122,125}, {127,525}, {599,5642}, {2777,10718}, {2871,3917}, {6054,10519}

= reflection of X(6793) in X(2)

= Tripolar centroid of X(2419)

= anti-centroidal-pedal of X(1297)

= [ 4.521334610152723, 2.17917383144425, 0.045235855452488 ]

 

 

César Lozada

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