How I stumbled upon a new (to me) construction of the inverse of a point

While explaining to a friend analyst that a theorem about circle inversion that he used could be proved with synthetic geometry, I stumbled upon a new to me construction of the inverse of a point with respect to a circle. In this snapshot I describe episodes from this discovery process, faithfully to how they played out in time, highlighting the main ways in which I used dynamic geometry as a research tool.