IMO 2000 Problem 6: Euclidean-geometry translation
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In triangle ABC, let I be the incenter, T₁,T₂,T₃ the incircle contact feet on BC, CA, AB, and H₁,H₂,H₃ the corresponding altitude feet. Reflect H₁,H₂ in T₁T₂ and H₂,H₃ in T₂T₃. Let Z be the intersection of the lines through the two reflected pairs. Prove IZ=IT₁.
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 target links the orthic triangle and incircle through reflected lines and a circle-membership conclusion.
What counts as progress: A derivation placing the reflected-line intersection Z on the incircle, with the stated contacts and reflections retained.
How the problem developed
International Mathematical Olympiad Published the original IMO 2000 Problem 6 geometry problem at the annual contest. The AG30 file later gives its specialized translated statement.
Trieu H. Trinh and coauthors Included a translated version in the 30-problem AlphaGeometry benchmark; the study reports 25/30 solved overall, which does not identify this task’s individual result.
- 2025 · Problem level benchmark [4]
Yuri Chervonyi and coauthors Figure 8 of the AlphaGeometry 2 study lists the original IMO 2000 Problem 6 as solved by AlphaGeometry 1; for Problem 2, the chart does not separate the translated 2a and 2b tasks.
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 2000 official problem statement booklet
- Google DeepMind, AlphaGeometry 30 translated problem statements
- AlphaGeometry study and benchmark results
- Chervonyi et al., Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2
- Trinh and collaborators: Solving olympiad geometry without human demonstrations
- AlphaGeometry source and proof artifacts
- DeepMind account of computer checking
- Precise translated question and assumptions
Website timeline
Publication dates and editorial changes.
AlphaGeometry solves 25 translated Olympiad geometry problems