The following discussion may be of interest on account of the way in which it leads to the axial and focal properties.
I. Let the locus of a point P such that A(CDEP) = B(CDEP), where A, B, C, D, E are fixed points, be called the conic (AB, CDE). Then it is clear that permuting A, B or C, D, E does not affect the conic, and that the conic passes through C, D, E, and also through A, B. (Give BP the four positions BC, BD, BE, BA and then give AP the position AB.)
and 