# Construction of Conics from Pascal's Theorem

Given five points on a conic, Pascal's theorem helps find arbitrarily many points on the same conic. Let, for example, points A, B, C, D, E are known to be located on a conic. If F is a sixth point on the conic then, according to Pascal's theorem, the intersections

The *converse* of Pascal's theorem, proved independently by W. Braikenridge and C. MacLaurin [Coxeter & Greitzer, p. 76] states that

If the three pairs of opposite sides of a hexagon meet at three collinear points, then the six vertices lie on a conic, which may degenerate into a pair of lines.

If F is not given and X is a random point in the plane, then define

The applet below illustrates this construction.

What if applet does not run? |

The derivative of this construction is the method of MacLaurin. To understand how it works, observe what happens when you drag point X. Point P remain fixed, Q slides along CD while R slides along BC. None of other points moves, except for F, which remains on the conic as expected.

The formulation of MacLaurin's construction begins with a variable triangle QRF whose side lines QR, FQ, FR pass through the fixed points P, A, E, while the vertices Q and R move along the lines CD and BC. The construction also fixes B and D - the intersection of AP with the second line and that of EP with the first. Under these constraints vertex F describes the conic that passes through A and E and the intersection C of the two given straight lines.

### References

- H. S. M. Coxeter, S. L. Greitzer,
*Geometry Revisited*, MAA, 1967

### Conics

- Conic Sections
- Conic Sections as Loci of Points
- Construction of Conics from Pascal's Theorem
- Cut the Cone
- Dynamic construction of ellipse and other curves
- Joachimsthal's Notations
- MacLaurin's Construction of Conics
- Newton's Construction of Conics
- Parallel Chords in Conics
- Theorem of Three Tangents to a Conic
- Three Parabolas with Common Directrix
- Butterflies in a Pencil of Conics
- Ellipse
- Parabola
- 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

### Pascal and Brianchon Theorems

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