  In this paper we study the most-demanding predicate for computing the
Euclidean Voronoi diagram of axes-aligned line segments, namely the Incircle
predicate. Our contribution is two-fold: firstly, we describe, in algorithmic
terms, how to compute the Incircle predicate for axes-aligned line segments,
and secondly we compute its algebraic degree. Our primary aim is to minimize
the algebraic degree, while, at the same time, taking into account the amount
of operations needed to compute our predicate of interest.
  In our predicate analysis we show that the Incircle predicate can be answered
by evaluating the signs of algebraic expressions of degree at most 6; this is
half the algebraic degree we get when we evaluate the Incircle predicate using
the current state-of-the-art approach. In the most demanding cases of our
predicate evaluation, we reduce the problem of answering the Incircle predicate
to the problem of computing the sign of the value of a linear polynomial (in
one variable), when evaluated at a known specific root of a quadratic
polynomial (again in one variable). Another important aspect of our approach is
that, from a geometric point of view, we answer the most difficult case of the
predicate via implicitly performing point locations on an appropriately defined
subdivision of the place induced by the Voronoi circle implicated in the
Incircle predicate.
