  The famous concyclicity theorem stated by John H. Conway is here reconsidered
by means of a parametrisation of the associated triangular configuration with
arbitrary triplets of real numbers ($\alpha$;$\beta$;$\gamma$). This theorem,
thus corresponding to the case ($\alpha$;$\beta$;$\gamma$)=(1;1;1), is
generalized while demonstrating that there always exist an infinite family of
such triplets which keeps unchanged the conclusion. The "anti-Conway"
configuration corresponding to the case ($\alpha$;$\beta$;$\gamma$)=(-1;-1;-1)
is also investigated : Xavier Dussau's theorem of concurrent lines is
redemonstrated and completed by another concyclicity theorem. It is also proved
that there exist in general a unique triplet
($\alpha$;$\beta$;$\gamma$)$\ne$(-1;-1;-1) which is a function of the sides of
the considered triangle and which keeps unchanged the conclusion of Dussau's
theorem.
