INTERNAL CENTER OF SIMILITUDE OF (O),(I)

Let ABC be a triangle and A'B'C' the cevian triangle of I.

Denote:

Ab = (IBC) /\ ((I,IB)-B)

ie the other than B intersection of the circumcircle of IBC and the circle centered at I with radius IB

Ac = (IBC) /\ ((I,IC)-C)

ie the other than C intersection of the circumcircle of IBC and the circle centered at I with radius IC

Similarly:

Bc, Ba and Ca, Cb

A" = BcBa /\ CaCb

B" = CaCb /\ AbAc

C" = AbAc /\ BcBa

A'B'C' and A"B"C" are perspective. The perspector S is on the OI line, the internal center of similitude of (O) and (I) = X(55).

Denote:

Ra = the radical axis of (IBC), (I,IA)

ie the radical axis of the circumcircle of IBC and the circle centered at I with radius IA.

Similarly Rb, Rc.

A* = AbAc /\ Ra

B* = BcBa /\ Rb

C* = CaCb /\ Rc

The triangles ABC, A*B*C* are perspective at a point P on the circumcircle of ABC.

The point is the center of the Jerabek hyperbola of the antipedal triangle of I (lying on the NPC of the antipedal of I = circumcircle of ABC)

Orthic triangle variation:

Internal center of similitude of circumcircle of orthic [=NPC of ABC] and incircle of orthic [=pedal circle of incenter of orthic = H of ABC] on the Euler line of ABC.

Perspector P lies on the NPC and is the center of the Jerabek hyperbola, X(125).

Antreas P. Hatzipolakis, 2 May 2013