Abstract: The Brouwer fixed point theorem states that if f is a continuous map of the n-ball Bn to itself, then there exists x such that f(x) = x. One usually proves this by saying "suppose not" and then appealing to degree or homology to get a contradiction. The machinery gives a proof of Borsuk-Ulam: if f maps Sn to En there is an x such that f(-x) = f(x) (i.e., some pair of antipodal points goes to the same point).
Francis Edward Su (Amer. Math. Monthly 104 (1997) p.855-859) gives an elementary geometric direct construction proving that Borsuk-Ulam implies Brouwer. Let f be a map of Bn to itself. We construct a map g of the n-sphere Sn into En, Euclidean n-space (which contains the n-ball). An x such that g(-x) = g(x) will give us a fixed point of f.
Some major machinery is needed to get Borsuk-Ulam directly, so this is not an elementary proof of Brouwer, a fact that the author does not make clear. However, the construction is nice and requires only the knowledge of n-dimensional geometry to be found in an elementary linear algebra course or a good calculus course.