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3817Isoxxx Conjugate

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• Jun 18, 2017

Dear all,

I defined the term system centers in message #2314

and found the third one X(1138) besides X(4) and X(74).

There are several topics about system centers.

Which center is the Feuerbach point?

A property of X1138

Concurrent lines with Neuberg cubic

Circular(Isotropic) Coordinate System

Selfcentric Centers

X(74)

Incenter/excenters X1,X1a,X1b,X1c form an X(4) system.

We can deduce isogonal conjugation from them.

In message #3816, I got the X(74) system X?,X?a,X?b,X?c.

Their barycentrics:(using Conway's notation)

X? - sqrt(a^2-2S_B S_C/S_A):sqrt(b^2-2S_C S_A/S_B):sqrt(c^2-2S_A S_B/S_C)

X?a - -sqrt(a^2-2S_B S_C/S_A):sqrt(b^2-2S_C S_A/S_B):sqrt(c^2-2S_A S_B/S_C)

X?b - sqrt(a^2-2S_B S_C/S_A):-sqrt(b^2-2S_C S_A/S_B):sqrt(c^2-2S_A S_B/S_C)

X?c - sqrt(a^2-2S_B S_C/S_A):sqrt(b^2-2S_C S_A/S_B):-sqrt(c^2-2S_A S_B/S_C)

They all lie on the polar circle.

As the polar circle, they may be imaginary.

They are missing in the ETC.

We can deduce isoxxx conjugation from them.

x:y:z -> (a^2-2S_B S_C/S_A)/x:(b^2-2S_C S_A/S_B)/y:(c^2-2S_A S_B/S_C)/z

Surpringly!  It has appeared in CTC already.

There are two pivotal isocubics K059=pK(X1990,X4)

and K543=pK(X1990,X5667) and two non-pivotal isocubics

K393=nK0+(X1990,X4) and X628=nK+(X1990,X27,X1).

The poles in CTC term are all X1990.  Please refer to

the attchment file for K059 and K543.

I hope this conjugation to be applied to more cubics.

Because they lie on the polar circle, I recommend the term

isopolar conjugate.

To be continued...

Best regards,

Tsihong Lau