Projective geometry · formal theorem
For six recorded nonzero tangent lines to a nonsingular real projective conic, the three joins of opposite consecutive-line intersections are concurrent.
Scope: For six recorded nonzero lines tangent to a nonsingular real projective conic, the three joins of opposite consecutive-line intersections are concurrent.
Exact formal proposition
theorem brianchonTheorem : BrianchonTheoremStatementProof route
The checked proof maps tangent lines of a nonsingular homogeneous conic to points on the adjugate dual conic, applies an algebraic Pascal collinearity theorem there, and uses dot–cross duality to return to concurrence of the three primal diagonals.
These stages follow the checked source and explain the mathematical route. They summarize the argument; they are not a visualization of Lean's internal proof term.
q(x)=xᵀQx; det(Q) ≠ 0
The proof begins with a symmetric homogeneous quadratic form whose matrix Q is nonsingular. This determinant hypothesis is exactly what makes the adjugate dual conic available.
quadNonsingularConicdetSymdualConic
dualQuad(Q t)=2 det(Q)·quad(t)=0
For each selected point t on the primal conic, Q t gives its tangent-line coordinates. The identity dualQuad(Q t)=2 det(Q) quad(t) puts all six tangent lines on the dual conic.
tangentLinedualConicdualConic_tangentLine_quadTangentToConic
Pascal dual intersections P₁,P₂,P₃ are collinear
The six dual tangent points satisfy the algebraic Pascal hypotheses. Intersecting the three prescribed pairs of opposite sides yields three dual points on one common line.
pascalTheorem_algebraicpascalPoint₁pascalPoint₂pascalPoint₃Collinear
det(P₁,P₂,P₃)=0 ⇒ D₁,D₂,D₃ concurrent
Three dual intersections lie on one line. In the primal plane, six tangent lines form a circumscribed hexagon whose three opposite-vertex joins meet at one point.
dot_cross_cyclicconcurrent_of_dual_pascallineThroughbrianchonTheoremThe theorem at a glance
Accessible transcript
dual Pascal collinearity ⇔ primal Brianchon concurrency
The adjugate matrix defines the dual conic, the tangent identity places the six dual points on it, algebraic Pascal makes three dual intersections collinear, and a dot–cross identity converts that line into the primal concurrency point.
PROJECTIVE GEOMETRY · CONIC DUALITY
BRIANCHON’S THEOREM
6 TANGENTS · 3 DIAGONALS · 1 POINT
A NONSINGULAR CONIC IN HOMOGENEOUS REAL COORDINATES
Six tangent lines form a circumscribed hexagon.
THE THREE OPPOSITE-VERTEX DIAGONALS ARE CONCURRENT
HOW THE PROOF MOVES
1 · PASS TO THE DUAL CONIC
Each tangent line becomes a point on the adjugate conic.
2 · USE THE ADJUGATE IDENTITY
On-conic tangent points map onto the dual conic.
3 · APPLY PASCAL IN DUAL COORDINATES
Three dual intersection points are collinear.
4 · RETURN TO THE ORIGINAL PLANE
The dot–cross identity turns dual collinearity into concurrency.
EXACT SCOPE
Nonsingular real conics in homogeneous coordinates.
No degenerate-conic claim. The superseded polar-image-only v0 statement is false.
Theorem schematic
six conic tangents ⇒ three opposite diagonals concurrent
Brianchon’s theorem sends a circumscribed hexagon around a nonsingular conic to one concurrency point formed by its three opposite-vertex diagonals. The formal general-position predicate is weaker than the generic distinct configuration shown.
Result boundary
For six recorded nonzero lines tangent to a nonsingular real projective conic, the three joins of opposite consecutive-line intersections are concurrent.
These AI-generated visuals explain the theorem and proof route; they are not proof evidence. Their publication review was completed separately from review of the formal result. The exact Lean proposition and checked source remain authoritative.
Line counts exclude blank lines; comments and documentation count. The total is the commit-pinned first-party Lean import closure; Mathlib and other third-party dependencies are excluded.
Continue the mathematics
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Review record
Lean checks the exact proof. Independent reviewers accepted evidence completeness, statement alignment, scope, and public wording; the visual review covers explanation only.
Independent review accepted the recorded build, exact declarations, unfinished-step scan, and axiom evidence.
The formal declaration was accepted against the named theorem and its exact variant.
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Independent review accepted the retained theorem explanation and source presentation. Generated media follows a separate review and promotion gate.
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A validated accepted-result record binds the four reviews to the checked formalization.