The concurrency may even be taken in projective way (the case when the support lines of the segments are parallel, and for that particular case the problem is a well-known application of Helly's theorem (mentioned in the Encyclopaedia Britannica), as sketched below). Consider the (dual) $2$-dimensional vector space $V$ of the lines of the plane. Define by $L_k$ the set of lines intersecting the segment $\Delta_k$, for $1\leq k\leq n$. The sets $L_k$ are convex in $V$, and by the givens, any three have a non-empty meet. By Helly's theorem, them all $n$ have a non-empty meet, and so any line in that meet intersects all $n$ segments.

Rembetophile Do you mean Wikipedia? Anyway, Wikipedia contains it http://en.wikipedia.org/wiki/Helly%27s_theorem

No, phile I mean the problem with segments on parallel support lines (vertical) is given as example of an application to Helly's theorem in the Encyclopaedia Britannica.