Three focus-sharing conics

define a quadrilateral of intersections of lines connecting the intersections of common tangents. Here, the black conic is the absolute conic. Again, F1 is the focus common to K2 and K3, and the intersection of one pair of tangents common to these two conics. The other pair of common tangents meet in T1. The six points turn olut to be the vertices of a quadrilateral.