Morley's other miracle: displaystyle 4^(p-1)equivpm (smallmatrix p-1 frac{p-1}{2} endsmallmatrix) pmod {p³}

Frank Morley is famous for his theorem concerning the angle trisectors of a triangle. This note gives an elementary proof of another result of Morley's, which relates the middle binomial coefficient to a certain power of two. The striking thing about Morley's congruence is that it is valid modulo the third power of the prime being considered.