`Compfiles/Imo2021P3.lean`

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 8 curated AI-written annotations and 3305 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned compfiles documentation, Pinned compfiles corpus index

1 declarations, 1 contain sorry, 0 declared as axioms

International Mathematical Olympiad 2021, Problem 3

Let D be an interior point of the acute triangle $ABC$ with AB > AC so that ∠DAB = ∠CAD. The point E on the segment AC satisfies ∠ADE = ∠BCD, the point F on the segment AB satisfies ∠FDA = ∠DBC, and the point X on the line AC satisfies CX = BX. Let O₁ and O₂ be the circumcenters of the triangles ADC and EXD, respectively. Prove that the lines BC, EF, and O₁O₂ are concurrent.

Imports (2)

Imported by 1 file

Compfiles.lean

problem Imo2021P3.imo2021_p3 IMO 2021 Problem 3: three concurrent lines (unsolved here) sorry line 40 source
L4DEL explanation
A geometry statement about a triangle with a special interior point D; the formalization is stated but its proof is the single word sorry.
1. Problem: in an acute triangle ABC with AB greater than AC, D is an interior point with angle DAB equal to angle CAD; E on segment AC and F on segment AB satisfy angle ADE = angle BCD and angle FDA = angle DBC; X lies on line AC with CX = BX; O1 and O2 are the circumcenters of triangles ADC and EXD. Prove that lines BC, EF and O1O2 are concurrent.
2. Formal conclusion: E is not F, O1 is not O2, and the three lines BC, EF, O1O2 have a common point.
3. Status: the proof body is sorry, so no solution is present. The file credits the statement as imported from IMOLean by jsm28.
Sources and attribution: Pinned Lean source (47b19bbfe5f3)
depends on sorry what this means
Statement uses: NormedAddCommGroup, InnerProductSpace, Real, MetricSpace, NormedAddTorsor, Fact, Eq, Nat, Module.finrank, OfNat.ofNat, AffineIndependent, Fin, Nat.succ, Matrix.vecCons, Matrix.vecEmpty, Affine.Simplex.AcuteAngled, Affine.Simplex.mk, LT.lt, Dist.dist, Membership.mem and 13 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

sorry: the Lean placeholder that lets a declaration compile without a proof; any theorem that uses it is not actually proved.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
problem imo2021_p3 {A B C D E F X O₁ O₂ : P}
    (affineIndependent_ABC : AffineIndependent ℝ ![A, B, C])
    (acuteAngled_ABC : (⟨_, affineIndependent_ABC⟩ : Triangle ℝ P).AcuteAngled)
    (AC_lt_AB : dist A C < dist A B)
    (D_mem_interior_ABC : D ∈ (⟨_, affineIndependent_ABC⟩ : Triangle ℝ P).interior)
    (angle_DAB_eq_angle_CAD : ∠ D A B = ∠ C A D) (wbtw_A_E_C : Wbtw ℝ A E C)
    (angle_ADE_eq_angle_BCD : ∠ A D E = ∠ B C D) (wbtw_A_F_B : Wbtw ℝ A F B)
    (angle_FDA_eq_angle_DBC : ∠ F D A = ∠ D B C) (X_mem_AC : X ∈ line[ℝ, A, C])
    (CX_eq_BX : dist C X = dist B X)
    (affineIndependent_ADC : AffineIndependent ℝ ![A, D, C])
    (O₁_eq_circumcenter_ADC :
      O₁ = (⟨_, affineIndependent_ADC⟩ : Triangle ℝ P).circumcenter)
…
```
Mentions: P