IMO 2008 Problem 6: Euclidean-geometry translation
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Let X,Y,Z,W lie on a circle with center O. Intersections of its tangents at (Z,X), (Z,W), (Y,W), (X,Y) give A,B,C,D. Take the incircles of ABC and ACD. Their two chosen common tangents meet at K. Prove K lies on the original circle.
This page concerns the specialized translated task used in the IMO-AG-30 benchmark. The study’s aggregate result does not establish whether this individual task was solved.
Why it matters and what counts as progress
Why it matters: The configuration links the original circumcircle, a circumscribed quadrilateral and two incircles through common tangents.
What counts as progress: A proof that the common-tangent intersection satisfies OK=OX, preserving the benchmark’s tangent choices and nondegenerate construction.
How the problem developed
International Mathematical Olympiad The official archive lists this as IMO 2008 P6; it is one of the contest problems later encoded in the specialized IMO-AG-30 geometry language.
Trieu H. Trinh, Yuhuai Wu, Quoc V. Le, He He, and Thang Luong (AlphaGeometry) The Nature paper identifies this as unsolved by AlphaGeometry. It also presents a human proof and reports that supplying the human auxiliary constructions still did not produce a system proof.
Yuri Chervonyi, Trieu H. Trinh, Miroslav Olšák, Xiaomeng Yang, Hoang H. Nguyen, Marcelo Menegali, Junehyuk Jung, Junsu Kim, Vikas Verma, Quoc V. Le, and Thang Luong (AlphaGeometry2) The AlphaGeometry2 paper reports solving all 30 statements in the original IMO-AG-30 benchmark, which includes this formalized target. This is a system result on the benchmark encoding, not an independent review of the original contest wording.
AI contribution
AlphaGeometry solves 25 translated Olympiad geometry problems
AlphaGeometry solved 25/30 specialized IMO-AG-30 classical Euclidean geometry translations using learned auxiliary-point constructions plus a symbolic deduction and algebra engine; the paper does not claim all natural-language IMO geometry conditions are represented.
Sources
- IMO 2008 Problems (official archive)
- Trinh et al., Solving olympiad geometry without human demonstrations
- AlphaGeometry IMO-AG-30 formalized problem set
- Chervonyi et al., Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2
- AlphaGeometry2 paper, arXiv record
- Trinh and collaborators: Solving olympiad geometry without human demonstrations
- AlphaGeometry source and proof artifacts
- DeepMind account of computer checking
- Precise translated question and assumptions
- AlphaGeometry study and benchmark results
Website timeline
Publication dates and editorial changes.
AlphaGeometry solves 25 translated Olympiad geometry problems