Pascal's Theorem, Homogeneous Coordinates
The theorem states that if a hexagon is inscribed in a conic, then the three points at which the pairs of opposite sides meet, lie on a straight line.
The theorem has been demonstrated on a circle and ellipse and proved based on Chasles' theorem and also on Menelaus' theorem.
Below we give an additional proof that makes use of homogeneous coordinates. This particular proof comes from a book by E. A. Maxwell, popular half a century ago. The proof reached me through the good services of Professors Hubert Shutrick and his friend Gordon Walsh.
The proof refers to the diagram and notational conventions of the previous proofs. Pascal's mystic hexagon ABCDEF is inscribed into a conic with an equation in homogeneous coordinates
f(x, y, z) = ax² + by² + cz² + fxy + gxz + hyz = 0.
Choose the coordinates so that
f(A) = f(1, 0, 0) = ax² = 0.
So that, with this choice of coordinates, a = 0. Similarly, we find that b = c = 0. After the division by xyz, the equation of the conic becomes
(*) | f / x + g / y + h / z = 0. |
Denote D, F and B by (x_{1}, y_{1}, z_{1}),
EF is x/x_{2} = y/y_{2}, BC is x/x_{3} = z/z_{3} so that R is ( 1 , y_{2}/x_{2}, z_{3}/x_{3});
AB is y/y_{3} = z/z_{3}, DE is x/x_{1} = y/y_{1} so that P is (x_{1}/y_{1}, 1 , z_{3}/y_{3});
AF is y/y_{2} = z/z_{2}, CD is x/x_{1} = z/z_{1} so that Q is (x_{1}/z_{1}, y_{2}/z_{2}, 1 ).
P, Q and R are collinear if the determinant of the matrix
1 | y_{2}/x_{2} | z_{3}/x_{3} | ||||||
x_{1}/y_{1} | 1 | z_{3}/y_{3} | ||||||
x_{1}/z_{1} | y_{2}/z_{2} | 1 |
is zero. After you divide the first column by x_{1}, second by y_{2} and third by z_{3}, you see that, by (*), f times row 1 plus g times row 2 plus h times row 3 is zero. Which means that the determinant is indeed 0 and the points P, Q, R are collinear. Q.E.D.
References
- E. A. Maxwell, Methods of Plane Projective Geometry Based on the Use of General Homogeneous Coordinates, Cambridge, 1960
Pascal and Brianchon Theorems
- Pascal's Theorem
- Pascal in Ellipse
- Pascal's Theorem, Homogeneous Coordinates
- Projective Proof of Pascal's Theorem
- Pascal Lines: Steiner and Kirkman Theorems
- Brianchon's theorem
- Brianchon in Ellipse
- The Mirror Property of Altitudes via Pascal's Hexagram
- Pappus' Theorem
- Pencils of Cubics
- Three Tangents, Three Chords in Ellipse
- MacLaurin's Construction of Conics
- Pascal in a Cyclic Quadrilateral
- Parallel Chords
- Parallel Chords in Ellipse
- Construction of Conics from Pascal's Theorem
- Pascal: Necessary and Sufficient
- Diameters and Chords
- Chasing Angles in Pascal's Hexagon
- Two Triangles Inscribed in a Conic
- Two Triangles Inscribed in a Conic - with Solution
- Two Pascals Merge into One
- Surprise: Right Angle in Circle
Chasles' Theorem
- Chasles' Theorem
- Chasles' Theorem, a Simple Proof
- Pascal's Theorem, Homogeneous Coordinates
- Chasles' Theorem, a Proof
- Projective Proof of Pascal's Theorem
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