TriangleConicsIntersections

Triangle-conics intersections

Consider two conics {c, c'} circumscribing triangle ABC with corresponding perspectors {P,Q}. Their fourth intersection point S can be constructed as follows:
[1] Define the intersection point R of line PQ with side-line BC.
[2] Take the harmonic conjugate R* of R with respect to {B,C}.
[3] Line AR* passes through the fourth intersection point S of the two conics.

The property follows from Maclaurin's generation of the conics discussed in ConicsMaclaurin2.html . There it is shown that taking the harmonic associate P* of the pivot P of the conic we can describe it through intersection-points of variable lines. For this, arbitrary lines through P* are drawn which cut sides {AB, AC} at points {B',C'} correspondingly. The conic is generated by the intersection-points of lines {BC',CB'}.
Applying this idea to the harmonic conjugates {P*,Q*} of the two pivots {P,Q} and the corresponding line P*Q* we obtain the intersection point S of lines {BC',CB'} belonging to both conics.
The rest is an easy consequence of this construction of S. By their definition lines {AR, PQ, BC, B'C'} make a harmonic bundle. It follows that the pole of line B'C' is the same point R* with respect to both conics. It follows also that line R*S passes through A and R* is harmonic conjugate to R with respect to {B,C}.

Remark-1 All conics whose perspector K lies on line PQ pass through S. The conic with perspector uP+vQ is uc+vc'. These are all the members of the family of conics having in common the four points {A,B,C,S}. There are three degenerate members in this family, consisting of the line-pairs {B'A,B'C}, {C'A,C'B} and {R*A,R*B}. They correspond to positions of K on the sides of the triangle i.e. the three intersection points of line PQ with the sides of the triangle.
Remark-2 Taking as one of the circumconics (c) to be the circumcircle one has an easy method to determine the so-called fourth intersection point of the conic (c'). A particular case is the Steiner point whose figure is shown in SteinerPoint.html .

Triangle-conics intersections

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