Even more elementary than Morley's theorem is Napoleon's.

Take any triangle, and construct an equilateral triangle on each of its sides. Then their midpoints form an equilateral triangle, too.

I don't know a striking application of the theorem itself. But from the proof, you can also conclude that if the point $D$ inside the triangle $\Delta ABC$ minimises $d(A,\cdot)+d(B,\cdot)+d(C,\cdot)$, then the line segments $AD$, $BD$ and $CD$ meet at angles $\frac{2\pi}3$ (However, here you need an extra assumption that all angles of $\Delta ABC$ are less than $\frac{2\pi}3$, so this does not count for this question).