CONCYCLIC POINTS. LOCUS

A CIRCLE:

Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3 the NPC centers of IB'C', IC'A', IA'B', resp.

The points I, N1,N2,N3 are concyclic.

ETC X(5453)

Center of the circle?

LOCUS:

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and N1, N2, N3 the NPC centers of PB'C', PC'A', PA'B', resp.

Which is the locus of P such that the points P,N1,N2,N3 are concyclic ?

Antreas P. Hatzipolakis, 17 April 2013, Hyacinthos #21970

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The center X of the circle has trilinears:

2*cos(A)+4*sin(3*A/2)*cos(B/2-C/2)+ cos(B-C)+2 : :

ETC search: 2.387773069046934.., 2.38593313995937.., 0.886815506990847..

X = Midpoint of X(I),X(J) for these (I,J): (1,500)

X lies on line X(I),X(J) for these (I,J):

(1,30), (3,81), (5,581), (21,323), (58,5428), (140,3216), (186,2906), (386,549), (511,1385), (550,991), (1154,2646), (2771,3743)

Cιsar Lozada Hyacinthos #21972

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A related locus:

Q such that Q and the NPCs of BCQ, CAQ, ABQ are concyclic. This would include X(13), X(14), the bicentric pair PU(5), the circumcircle intercepts of line X(5)X(523), and the point Qi (of ABC) such that I = Qi of the cevian triangle of I.

Randy Hutson, Hyacinthos #21977

This is the tricircular sextic: Q014 - 4 S^2 x y z (x + y + z) (c^2 x y + b^2 x z + a^2 y z) where S = twice the area of ABC. (tricircular = the circular points are triple points of the curve).

Francisco Javier Hyacinthos #21981

.... the point Qi on this curve is the point which is the incenter of its anticevian triangle, and has ETC search value 1.999434154060428. Coordinates? Its isogonal conjugate Qi* is the point which is the nine-point center of its pedal triangle (ETC search 1.142779079509848). The line QiQi* passes through X(5).

Randy Hutson, Hyacinthos #21982

Does this locus contain any ETC centers or bicentric pairs besides X(13), X(14), and PU(5)? Are the circumcircle intercepts of line X(5)X(523) triangle centers or another bicentric pair?

Randy Hutson, Hyacinthos #21983