A CYCLOLOGIC THEOREM RELATED TO EXCENTRAL TRIANGLE.

[APH]

Excentral version

Let ABC be a triangle, IaIbIc the excentral triangle and P a point..

Denote

Pa, Pb, Pc = same to P points of IaBC, IbCA, IcAB, resp.

ABC, PaPbPc are circumcyclologic

Cyclologic center (ABC, PaPbPc) = Q = ? (on the circumcircle of ABC)
Cyclologic center (PaPbPc, ABC) = Q* = ? (on the circumcircle of PaPbPc)

[Ercole Suppa]

1. P on the Euler Line:

Q = X(100)
Locus of Q* as P moves on the Euler line: K086

2. P on the Brocard axis:

Q = X(101)
Locus of Q* as P moves on the Brocard axis: K040

[APH]

Let's see the Q's. The class of the cubics is a subject of Bernard Gibert.
1. Euler line
Q = X(100) = Reflection point of IO = X(1)X(3) line = Reflection point of Euler line of INTOUCH triangle (pedal triangle of I).

2. Brocard axis
Q = X(101) = Reflection point of X(1)X(7) line = Reflection point of Brocard axis of INTOUCH triangle (pedal triangle of I).