Thales_Theorem

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

If A and B are diametrically opposite points on a circle and C is another point on the circle, then angle ACB is a right angle.

Why this node was retired

The posted statement is

theorem Thales_Theorem (A B C : ℝ × ℝ) (center : ℝ × ℝ) (r : ℝ) (hr : 0 < r) (hA : dist A center = r) (hB : dist B center = r) (hC : dist C center = r) (hdiam : A.1 + B.1 = 2 * center.1 ∧ A.2 + B.2 = 2 * center.2) (hne : C ≠ A ∧ C ≠ B) : (C.1 - A.1) * (C.1 - B.1) + (C.2 - A.2) * (C.2 - B.2) = 0 := by sorry

The circle is defined in the product sup metric, but the conclusion uses Euclidean orthogonality.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Use the Euclidean norm and dot product consistently: for antipodal A,B on a Euclidean circle and C a different point on that circle, prove (A−C)·(B−C)=0.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.