  Tarski initiated a logic-based approach to formal geometry that studies
first-order structures with a ternary betweenness relation \beta, and a
quaternary equidistance relation \equiv. Tarski established, inter alia, that
the first-order (FO) theory of (R^2,\beta,\equiv) is decidable. Aiello and van
Benthem (2002) conjectured that the FO-theory of expansions of (R^2,\beta) with
unary predicates is decidable. We refute this conjecture by showing that for
all n>1, the FO-theory of the class of expansions of (R^2,\beta) with just one
unary predicate is not even arithmetical. We also define a natural and
comprehensive class C of geometric structures (T,\beta), and show that for each
structure (T,\beta) in C, the FO-theory of the class of expansions of (T,\beta)
with a single unary predicate is undecidable. We then consider classes of
expansions of structures (T,\beta) with a restricted unary predicate, for
example a finite predicate, and establish a variety of related undecidability
results. In addition to decidability questions, we briefly study the
expressivities of universal MSO and weak universal MSO over expansions of
(R^n,\beta). While the logics are incomparable in general, over expansions of
(R^n,\beta), formulae of weak universal MSO translate into equivalent formulae
of universal MSO.
