Hello, Narek!

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

Given a point $\displaystyle A$ interior to angle $\displaystyle POQ. $

Find the center and radius of the circle which passes through $\displaystyle A$

and is tangent to the sides of the angle. Code:

P
/
/ * * *
/* *
E * *
* *
/ *
/* * C *
/ * o *
/ * * * *
/ * *
/ * * * *
/ * * o A
/ * * *
O * - - - - - - - * * * - - - - Q
D

The center lies on the angle bisector $\displaystyle OC\!:\;\;CD = CE.$

The center is also equidistant from point $\displaystyle A$ (focus) and a line $\displaystyle OQ$ (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.