HYACINTHOS 25625

[Antreas P. Hatzipolaks]:

Let ABC be a triangle, A'B'C' the cevian triangle of H and L a line.

Denote:

A* = L Intersection AA'

B* = L Intersection BB'

C* = L Intersection CC'

1. Which is the envelope of L such that the NPC centers Na, Nb, Nc of AOA*, BOB*, COC*, resp. are collinear?

If L passes through O, then the the NPC centers Na, Nb, Nc of AOA*, BOB*, COC*, resp. are collinear.

2. Which is the envelope of the line NaNbNc as P moves around O ?

 

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Denote::

A" = L intersection BC

B" = L intersection CA

C" = L intersection AB

The envelope of the line L such that the circumcircles of AOA", BOB", COC" are coaxial is O (that is the circle with center O and radius 0, that is the point O) U the Kiepert parabola.

Reference

 

Is it true that:

The NPC centers of AOA*, BOB*, COC* are collinear <===> The Circumcircles of AOA", BOB", COC" are coaxial ?

(that is, the envelope in both case is the same ?)

[César Lozada]:

 

2) Conic with barycentrics equation:

CyclicSum[(21*R^4-4*(3*SA+5* SW)*R^2-8*S^2-4*SA^2+8*SA*SW+ 4*SW^2)*x^2+2*(21*R^4-(26*SA- 6*SW)*R^2+4*S^2+8*SA*SW-4*SW^ 2)*y*z ] = 0 (No ETC centers on it)

 

Center=X(140)

Perspector:

PP = 1/(2*(b^2+c^2)*a^6-3*(2*b^4+b^ 2*c^2+2*c^4)*a^4+(b^2+c^2)*(6* b^4-11*b^2*c^2+6*c^4)*a^2-2*( b^4-b^2*c^2+c^4)*(b^2-c^2)^2) : : (barycentrics)

= on line: {6748,7577}

= isogonal conjugate of {3,143}∩{51,381}

= [ 2.464425606837616, 2.80192351193484, 0.563444078181358 ]

 

César Lozada