Morley's other miracle: ${\displaystyle 4^{p-1}\equiv \pm (\text{smallmatrix}p-1\frac{p-1}{2}\text{endsmallmatrix})(\mathrm{m}\mathrm{o}\mathrm{d}{p}^3)}$

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.