### MacLaurin's Construction of Conics

Let ABC be a variable triangle which is such that the vertices B and C respectively move on two given lines *l* and *m*, and the sides BC, CA, and AB respectively pass through given points U, V, and W. Then the locus of the vertex A is a conic. If points U, V, W, are collinear then the locus degenerates into a straight line concurrent with *l* and *m*.

What if applet does not run? |

### 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
- D. Pedoe,
*Geometry: A Comprehensive Course*, Dover, 1970. p. 328 - G. Salmon,
*Treatise on Conic Sections*, Chelsea Pub, 6e, 1960, pp. 247-248 - 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
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Copyright © 1996-2018 Alexander Bogomolny

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander BogomolnyLet ABC be a variable triangle which is such that the vertices B and C respectively move on two given lines

*l*and*m*, and the sides BC, CA, and AB respectively pass through given points U, V, and W. Then the locus of the vertex A is a conic. If points U, V, W, are collinear then the locus degenerates into a straight line concurrent with*l*and*m*.What if applet does not run? ### Proof

The theorem, as stated, is a consequence of Pascal's theorem and is due to C. MacLaurin (1698-1746).

A reference to the construction of conics implied by Pascal's theorem requires a change of notations: let FQR be a variable triangle which is such that the vertices Q and R respectively move on two given lines CD and BC, and the sides QR, FR, and FQ respectively pass through given points P, E, and A. Then the locus of the vertex F is a conic. If points A, E, P, are collinear then the locus degenerates into a straight line through concurrent with CD and BC.

### References

### Conics

- D. Pedoe,

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