Sasha,
You might have to clarify the question. Does the family of all tangents to a circle qualify as a family you might be interested in? Each tangent has one parallel tangent, and the rest intersect it. For example, the family of tangents to the unit circle x^{2} + y^{2} = 1 have the form ax + by = 1, for numbers a, b that satisfy a^{2} + b^{2} = 1.
Chris
