I can't prove (but Cabri seems to confirm) the following conjecture about
any circle cutting the sides of ABC in six points, such as the point Ab on
BC nearest to B, and so on:
Let AbBa and AcCa meet at X (similarly for Y,Z) and AX meet BC at D
(similarly for E,F).
Let BaCb and CaBc meet at X' (similarly for Y',Z') and AX' meet BaCa at A'
(similarly for B',C').
Then A'D, B'E, C'F are concurrent (?).
For circles concentric with the incircle (recently discussed by Floor),
A' is the midpoint of the chord BaCa and D is the point of contact of the
incircle, etc. The concurrence in this case was proved in a direct way
recently by Floor and Antreas. Its locus is an arc of the Kiepert
hyperbola of the intouch triangle DEF. It may be worth noting that these
"expanded incircles" are a particular case of a coaxaloid system associated
with ABC for which AX,BY,CZ will be concurrent at a fixed point of the
system (J Third, 1899) - in the particular case this point is the Gergonne
point of ABC.