**Abstract:** The Brouwer fixed point theorem states that if *f* is a continuous map of the *n*-ball *B*^{n} 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 *S*^{n} to *E*^{n} 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 *B*^{n} to itself. We construct a map *g* of the *n*-sphere *S*^{n} into *E*^{n}, 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.