Poncelet triangles with fixed barycenter: choosing the caustic

Let T be a family of Poncelet triangles inscribed in an ellipse E and circumscribing some caustic E', for which the barycenter is stationary. Let C=(x,y) be a desired center for the caustic. The video shows a construction (based on a MacBeath pre-image) that allows one to visualize E' over C.