Side-side-angle with fixed angles and the cyclic quadrilateral (Q6617356)

Continuing a line of investigation started by the first and last author of this paper [Electron. Res. Arch., 31, 3, 1271--1286 (2023; \url{doi:10.3934/era.2023065})], it is shown that:\N\NIf \(ABC\) and \(ABC'\) are two non-congruent triangles with \(AB=c\), \(\widehat{ACB}=\widehat{AC'B}=\gamma\), \(AC\equiv AC'= b\), with \(b>c\), with \(C\) and \(C'\) in the same half-plane determined by \(AB\), with \(\gamma\) and \(c\) fixed, but \(b\) variable, then the locus of the intersection points \(M\) of the lines \(BC\) and \(AC'\) and \(\overline{M}\) of the lines \(AC\) and \(BC'\), is a certain normal hyperbola.\N\NA projective generalization is also provided.\N\NThe proofs are computational. \texttt{Maple} was used ``to precisely validate some highly complex computations''. The figures were drawn with \texttt{GeoGebra}, which was also used to ``visually verify'' the assumptions.