MUn satellite is a tangent circle externally

MUn satellite is a circle tangent externally to the circumscribed circle of the obtuse triangle ∆ABC. A satellite is inscribed in each of the CAB, ABC and BCA angles. Show that the 6 points resulting from the circle-radius tangence reside on a hyperbole. (A circle is written in the XYZ angle if it is tangent to the YX and YZ rays.)
My try

$A_B=\mathrm{Bar}\left[\begin{array}{cc} A&B\-t_\alpha & t_\alpha + t_\gamma\end{array}\right]\ \ ;\ \ A_C=\mathrm{Bar}\left[\begin{array}{cc} A&C\-t_\alpha & t_\alpha + t_\beta\end{array}\right]\ \ ;\ \ B_A=\mathrm{Bar}\left[\begin{array}{cc} A&B\t_\beta + t_\gamma & -t_\beta\end{array}\right]\ \ ;\ \ \cdots$ with $tγ=tan⁡(γ/2)t_\alpha = \tan(\alpha/2)\ ;\ t_\beta = \tan(\beta/2)\ ;\ t_\gamma = \tan(\gamma/2)$

\And the 6 points are actually on the same conical (of a relatively simple barycentric equation) of which I have not yet looked at in detail the characteristics according to the angles.