Extended Data Fig. 4: Side-by-side comparison of human proof and AlphaGeometry proof for the IMO 2019 P2.

This is one out of five unsolved problems by AlphaGeometry. Left, the human solution uses both auxiliary constructions and barycentric coordinates. With a well-chosen coordinate system, a solution becomes available through advanced algebraic manipulation. Right, AlphaGeometry solution when provided with the ground-truth auxiliary construction for a synthetic proof. This auxiliary construction can be found quickly with the knowledge of Reim’s theorem, which is not included in the deduction rule list used by the symbolic engine during synthetic data generation. Including such high-level theorems into the synthetic data generation can greatly improve the coverage of synthetic data and thus improve auxiliary construction capability. Further, higher-level steps using Reim’s theorem also cut down the current proof length by a factor of 3.