We address a number of extremal point query problems when P is a set of n points in R d , d ⩾ 3 a constant, including the computation of the farthest point from a query line and the computation of the farthest point from each of the lines spanned by the points in P. In R 3 , we give a data structure of size O ( n 1 + ɛ ) , that can be constructed in O ( n 1 + ɛ ) time and can report the farthest point of P from a query line segment in O ( n 2 / 3 + ɛ ) time, where ɛ > 0 is an arbitrarily small constant. Applications of our results also include: (1) Sub-cubic time algorithms for fitting a polygonal chain through an indexed set of points in R d , d ⩾ 3 a constant, and (2) A sub-quadratic time and space algorithm that, given P and an anchor point q, computes the minimum (maximum) area triangle defined by q with P ∖ { q } .