Dear Antreas

Particular cases of the lines L:

Let ABC be a triangle, P a point, La the cevian AP, A'B'C' the pedal

triangle of P and A"B"C" the circumcevian triangle of P wrt A'B'C'.

Let A*,B*,C* be the orth. projections of A,B,C on La (A*=A).

The lines A*A",B*B",C*C" are concurrent in Xa, on pedal circle of the P.

Similary construct Xb, Xc (when making the lines BP and CP) then two triangles ABC and XaXbXc are perspective.

If P = (u : v : w) in barycentrics then the perspector is

Q= u/[u(b^2(u+v-w)+c^2(u-v+w)) - a^2(u^2+u(v+w)+2vw)]:... : ...

A first pair examples (P,Q):

(X(1), X(8)); (X(2),(1/(b^2+c^2-5a^2):...:...) ); (X(3),X(68)); (X(4),X(4));...

Best regards,

Angel M.

--- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis wrote:

>

> Let ABC be a triangle, L a line, P a point on L, A'B'C' the pedal

> triangle of P and A"B"C" the circumcevian triangle of P wrt A'B'C'.

> Let A*,B*,C* be the orth. projections of A,B,C on L.

>

> The lines A*A",B*B",C*C" are concurrent ( I think we have

> already discussed this here. Anyway.... I do not have time for searching...).

> If the point P is fixed and the line L moves around

> P, the locus of the point of concurrence is the pedal circle of P.

> [in what cases we get the Feuerbach point as point of concurrence?]

> If the line L is fixed and P moves on L, which is the locus of

> the point of concurrence?

>

> APH

>