- Dear friends,

I am new in using trilinears. As an exercise in learning trilinears I

invented following construction.

Let P = x : y : z, A'B'C' be P-cevian tringle of ABC. Let LA be the radical

axis of circles (AC'C) and (AB'B). Similarly we define lines LB and LC. It's

not difficult to prove that LA, LB and LC concur in P' = a/(by+cz):

b/(ax+cz): c/(ax+by)= sinA/(ysinB+zsinC): ...

We have mapping P -> P'. By this mapping, for example, X(1)-> X(58),

X(2)->X(6), X(3)-> X(54), X(4)-> X(4), X(6)-> X(251), X(8)-> X(1), X(20)->

X(3), X(63)-> X(284), X(144)-> X(55), X(192)-> X(31), X(193)-> X(25),

X(329)-> X(9),...

What are the following centers:

X{7}-> a^2 - (b - c)^2 : b^2 - (c - a)^2 : c^2 - (a - b)^2 = sinA/[tan(B/2)

+ tan(C/2)] : ...

X(10) -> a/(2a + b + c) : b/(2b + c+ a) : c/(2c + a + c) ?

We can obtain more cenetrs (or new construction for known centers).

Best regards,

Tatiana - Dear Tatiana,

[TE]: Let P = x : y : z, A'B'C' be P-cevian tringle of ABC. Let LA

be the radical axis of circles (AC'C) and (AB'B). Similarly we define

lines LB and LC. It's not difficult to prove that LA, LB and LC

concur in

P' = a/(by+cz): b/(ax+cz): c/(ax+by)

= sinA/(ysinB+zsinC): ...

We have mapping P -> P'.

*** This is the isogonal conjugate of the inferior of P. The inferor

of P is the point dividing PG externally in the ratio 3:-1.

By this mapping, for example, X(1)-> X(58),> X(2)->X(6), X(3)-> X(54), X(4)-> X(4), X(6)-> X(251), X(8)-> X(1), X

(20)->

> X(3), X(63)-> X(284), X(144)-> X(55), X(192)-> X(31), X(193)-> X

(25),

> X(329)-> X(9),...

= sinA/[tan(B/2) + tan(C/2)] : ...

> What are the following centers:

> X{7}-> a^2 - (b - c)^2 : b^2 - (c - a)^2 : c^2 - (a - b)^2

***This is X(57). It is a point on the line OI, and is the

intersection of the the three lines each joining an excenter to the

point of tangency of the incenter with the corresponding side.

> X(10) -> a/(2a + b + c) : b/(2b + c+ a) : c/(2c + a + c) ?

***This point is not in ETC.

> We can obtain more cenetrs (or new construction for known centers).

Best regards,

Sincerely,

Paul