Theorem: in any triangle ABC, the points of intersection of adjacent angle trisectors (the trisector of angle A nearest AB meeting the trisector of angle B nearest AB, etc.) form an equilateral triangle — the 'first Morley triangle'. Each triangle actually has 18 Morley triangles from different trisector combinations.
Morley discovered it around 1874 while studying higher-plane curves but did not publish until 1899 in the Transactions of the American Mathematical Society. The theorem cannot be proven by straightedge-and-compass construction (trisectors themselves are generally non-constructible), making it a bridge between classical and algebraic geometry. Conway called it 'the best theorem in elementary mathematics' and produced several proofs including one from complex coordinates in a single line.
First published proof: Frank Morley, 1899. Dozens of proofs since span synthetic geometry (via Carnot), complex analysis (Conway's one-liner), algebraic manipulation (via Ceva's theorem generalised to trisectors), and even projective approaches. The equilateral triangle persists under affine transformation of the parent triangle, connecting to the theory of cubic curves. A perfect example of a theorem that is easy to STATE, hard to BELIEVE, and surprisingly DEEP to prove.