Extended Data Fig. 5: Human proof for the IMO 2008 P6.

This is an unsolved problem by AlphaGeometry and also the hardest one among all 30 problems, with an average human score of only 0.28/7. This human proof uses four auxiliary constructions (diameters of circles W1 and W2) and high-level theorems such as the Pitot theorem and the notion of homothety. These high-level concepts are not available to our current version of the symbolic deduction engine both during synthetic data generation and proof search. Supplying AlphaGeometry with the auxiliary constructions used in this human proof also does not yield any solution. There is also no guarantee that a synthetic solution exists for AlphaGeometry, across all possible auxiliary constructions, without enhancing its symbolic deduction with more powerful rules. Again, this suggests that enhancing the symbolic engine with more powerful tools that IMO contestants are trained to use can improve both the synthetic data and the test-time performance of AlphaGeometry.