I have found this result by exploring for new problems.
If three parabolas share a common directrix and each pair intersect each other in two points, then, the lines joining the two intersection points of each pair of parabolas are concurrent.
The proof is quite simple, so my question is :
Has anyone seen this before? Any reference?
I found a useful reference and an interesting theorem called the Three Conics Theorem which is similar to yours, but not the same. Here is a link: http://mathworld.wolfram.com/ThreeConicsTheorem.html , and another reference: "The Seven Circles Theorem and other new theorems" by C.J.A. Evelyn, G.B. Money-Coutts, and J.A. Tyrrell (1974).
As you can see , the three conics intersect at I and J and intersect in pairs at (Q1,P1) , (Q2,P2) , (Q3,P3) . The three segments, Q1P1, Q2P2 , Q3P3 are concurrent at X. This picture is taken from the book mentioned above, (there is a similar diagram at mathworld).