[1] f maps the circles k

[2] f maps the euler circle (passing through D, E, F and the middles A', B', C' of HA, HB, HC respectively) to the circumcircle of ABC.

[3] f maps the incircle (g) of ABC to circle (d) which is simultaneously tangent to k

[4] f maps the

[1] Follows immediately from elementary properties of (anti)-inversions. In fact, f maps circle k

[2] The relation FH*HC = r

The rest follows from the fact that the incircle is simultaneously tangent to the sides and the Euler-circle (Feuerbach's theorem), hence by the anti-inversion (f) the incircle (g) maps to a circle (d) simultaneously tangent to k

Notice that circle (h) was met in the discussion of

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