- Let ABC be a triangle, L a line passing through H, intersecting

BC,CA,AB at A',B',C', resp.

Let La, Lb, Lc be the reflections of L in BC,CA,AB, resp. (concurrent

on the circumcircle).

Let Ab, Ac be the orth. projections of A' on Lb,Lc, resp. Similarly

Bc,Ba and Ca,Cb.

The circumcircles of A'AbAc, B'BcBa, C'CaCb are coaxial and their NPC

centers are

collinear. Which is this line ? Special case: L = Euler line.

APH - Antreas Hatzipolakis wrote:Old Theorem (*)

Let ABC be a triangle, P a point and A'B'C' the cevian triangle of P.

Denote:

P' = the Miquel point wrt triangle A'B'C'

(ie the point of concurrence of the circumcircles ofAB'C', BC'A', CA'B')A" = the other than A' intersection of BC and the circumcircle of

P'AA'. Similarly B", C".

The points A", B", C" are collinear.

Call the line A"B"C" as Lp(Note: The circumcircles of P'AA', PBB', PCC' are coaxial

The other than P' point of concurrence lies on Lp)

Some natural questions:1. Which line is Lp ? (ie whose point the trilinear polar is it)

2. Which is the locus of P such that Lp is parallel or perpendicularto a given line ? The Euler line for example.3. Which is the locus of P such that Lp passes through a given point?

O, G, N ..... for examples4. Which is the envelope of Lp as P moves on a given line?

The Euer line for exampe?

(*) Reference:

J. d. math. elem., 1 nov. 1940, p. 22, problem #13023

aph**********************************************************Dear AntreasSome natural questions:1. Which line is Lp ? (ie whose point the trilinear polar is it)*** Lp is the polar of the isogonal conjugate of P wrtcircumcircle.2.1. Which is the locus of P such that Lp is parallel to a given line? The Euler line for example.*** If the line is px+qy+rz = 0, the locus of P is thecircumconic of perspector:(a^2((b^2-c^2)p-b^2q+c^2r):...:...).For Euler line, the perspector is X(3003) and the circumconicpasses through points X(4), X(110).2.2 Which is the locus of P such that Lp is perpendicular to a given line ? The Euler line for example.*** If the line is px+qy+rz = 0, the locus of P is thecircumconic of perspector:(a^2(b^2(q*SA+p*SB-c^2r)+c^2(r*SA+p*SC-b^2q)): ... : ...).For Euler line, the perspector is X(3003) and the circumconicis the Jerabek Hyperbola.3. Which is the locus of P such that Lp passes through a givenpoint? O, G, N ..... for examples*** If the given point is (u:v:w), the locus of P is the circumconic of the perspector:(a^2(c^2 v + b^2 w) : b^2(c^2 u + a^2 w) : c^2(b^2 u + a^2v)), the crosssum of (u:v:w) and X(6).4. Which is the envelope of Lp as P moves on a given line?The Euler line for exampe?*** If the line is px+qy+rz = 0, the envelope of Lp is:b^2c^2(b^2c^2(q-r)^2x^2+ 2a^4(p^2-pq-pr-qr)yz+ ciclic sum =0.For Euler line, this conic is a parabola through points X(647), X(924)With infinity point X(924).Best regards,Angel Montesdeoca