Abstract The starting point is a pair of a circle {D} D and ellipse {E} E, which is within {D} D, so that there is a triangle inscribed in {D} D and circumscribed about {E} E. According to the famous Poncelet porism, we know that there is an infinite family of such triangles, so we will call \D, E\ D, E a 3-Poncelet pair. Using isogonal conjugacy, we develop a geometric construction that produces new families of circles that also form a 3-Poncelet pair with {E} E. In fact, any circle that is the locus of isogonal conjugates of a fixed point with respect to the family of Poncelet triangles of the original 3-Poncelet pair \D, E\ D, E serves as an outer circle in a new 3-Poncelet pair with the same inner conic {E} E. In the process, two special pairs of circles emerge as guide marks.
Garcia et al. (Mon,) studied this question.