I've been playing with amazing Geogebra software and finally found a construction that perfectly fits the description of a conjugate. Apparently there is a plethora of remarkable properties associated with this one. It can be defined as the circumcircular conjugate with respect to the circumcenter of the triangle ABC. I am still hoping that it even might be a new one :)
However as I am only aware of the cyclocevian, isotomic, isogonal and isocircular conjugations and there are lots of others listed in the ETC it usually would be expected that this conjugate is not new and just a lesser known one.
This conjugation works as follows:
R is the circumcenter of the triangle ABC. D - is a random point. Draw a circumcircle for a triangle DRA. A1 is the intersection point of the circumcircle of ABC and this circle. Points B1 and C1 can be constructed in the exact same manner. Then, lines AA1, BB1, CC1, RD always intersect at some point D1. It can be shown that this operation can be applied in the reverse order. Choosing D1 as the starting point we can construct points B2, A2, C2 (Green circle is the circumcircle of the triangle RBD1) and lines D1R, BB2, AA2, CC2 intersect exactly at the initial point D. So that apparently it is indeed a conjugate 2. (This needs to be strictly proven though).
I decided to find the circumcircular conjugate of the incenter of the triangle ABC
With the help of this construction we can get points I1, J that can be well known triangle centres... (or perhaps completely new ones) And finally we get the remarkable point X that is always lying on the circumcircle of the triangle ABC! This point X is especially intriguing to me. Could you please help me to identify it?