Let (c) be a conic and ABC a triangle inscribed in it. Let D be the pole of side BC and {X,Y} be the intersection points of a line through D and sides {AB,AC} correspondingly. The intersection point P of lines {BX,CY} describes the conic (c). This generation of the conic is discussed in ConicsMaclaurin2.html .
Here is an additional remark concerning the various positions of A on the conic. For all such triangles ABC (having variable A on (c) and B, C fixed on (c)) the corresponding perspector of the conic is point S, harmonic conjugate to D with respect to (A,G), G being the intersection point of AD with BC.
Changing the position of A on (c) (but maintaining the basis BC) the corresponding perspectors of the conic with respect to triangle ABC lie on a conic (c') belonging to the bitangent family generated by (c) and line BC.
In fact it is easily seen that the conic is the image of the projectivity F, which fixes points {B,C,D} and maps D' to E. Here D' denotes the reflexion of D with respect to BC and E is such that D'E : EK = 2.
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Locus of perspectors ![[0_0]](ConicsMaclaurin3_files/ConicsMaclaurin3_0_0_0.png)
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![[1_0]](ConicsMaclaurin3_files/ConicsMaclaurin3_0_1_0.png)
![[1_1]](ConicsMaclaurin3_files/ConicsMaclaurin3_0_1_1.png)