The first main result of this paper establishes that any sufficiently large subset of a plane over the finite field F-q, namely any set E subset of F-q(2) of cardinality vertical bar E vertical bar > q, determines at least q-1/2 distinct areas of triangles. Moreover, one can find such triangles sharing a common base in E, and hence a common vertex. However, we stop short of being able to tell how "typical" an element of E such a vertex may be.
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