### 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.

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Proof ### Conics

• Ellipse
• Parabola 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.

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### 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

1. D. Pedoe, Geometry: A Comprehensive Course, Dover, 1970. p. 328
2. G. Salmon, Treatise on Conic Sections, Chelsea Pub, 6e, 1960, pp. 247-248 68945260