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.
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The 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 The superseded polar-image-only v0 statement is false.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.
Proof architecture
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.
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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
Cross products encode primal vertices and diagonal lines. The dot–cross identity identifies incidence with the dual collinearity determinant, so the three opposite-vertex diagonals share one point.
dot_cross_cyclicconcurrent_of_dual_pascallineThroughbrianchonTheoremExact formal proposition
theorem brianchonTheorem : BrianchonTheoremStatementResult boundary
For six recorded nonzero lines tangent to a nonsingular real projective conic, the three joins of opposite consecutive-line intersections are concurrent.
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Continue the mathematics
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