## Maclaurin like

Consider a conic c and three points {A,B,C} lying on a line M. Let also L be an arbitrary line. For a point P varying on the conic consider lines {PA,PB}, the intersection E of PB and L and finally the intersection Q of lines AP and CE.
Q describes a second conic as P varies on c.

The result can be stated in a way resembling the Maclaurin's construction of conics (see Maclaurin.html ):

If the triangle PEQ has the properties:
(i) Its side-lines pass through three fixed points {A,B,C} lying on a line M,
(ii) One vertex (E) moves on a fixed line L,
(iii) A second vertex (P) moves on a conic c,
then its third vertex moves also on a conic c'.

The proof follows easily by defining an appropriate projectivity F, and obtaining c' as the image F(c). The projectivity is uniquely defined by the properties:
1) F fixes all points of line L and also point A,
2) F maps point B to C.
It follows that line BE maps under F to line CE and point P maps to Q.

Remark F is the perspectivity with center A, axis L and homology coefficient k equal to the cross ratio (A,O,B,C).