Hello, Narek!

I don't have it solved yet, but I have an observation . . .

Code:

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The center lies on the angle bisector

The center is also equidistant from point (focus) and a line (directrix).

. . This determines a *parabola.*

The center is the intersection of this parabola and the angle bisector,

. . and there are **two** such points.

Maybe someone can use this knowledge . . . or not.