Unimodality of the philon's Line

The following result was prooved by G. Toussaint and B. Bhattacharya in 1990. They needed to prove this result for a paper on the shortest transversals. This problem consist of computing the shortest line segment that intersects a set of n given line segments or lines in the plane.

Let QOR be the angle in consideration, P the point we are interested in and let H be the intercept of the philon line on OQ and K be the intercept of the philon line on OR. Let q be the angle subtended by RKH as in Fig. 1. And let l denote the length of line segment HK. Let l(q ) denote l as a function of q where f =q 1 < q