intersections of 2 conics

Two inconics meet at four points. Often one of these is known to be centrally defined. For example the incircle and the Mandart inconic meet at the Feuerbach point. The H and tH inconics meet at the center of the Jerabek hyperbola.

General algebraic principles would suggest that if four points share a method of construction and one is central, the other group of three must also be central. In the two cases cited above the other three intersections are perspective to ABC.

Figure: Two inconics meet at four points, one (o) is centrally defined, so the other group of three (a, b, c) are central and in perspective to ABC.

Wilson Stothers has written about this circumstance on his web pages, which discussion I recommend (along with everything on his pages), but I will proceed with my own way of developing this topic.

1. Lines lx+my+nz = o and Lx+My+Nz = o intersect at (: nL – Nl :).

2.Circumconics l/x+m/y+n/z = o and L/x+M/y+N/z = o intersect at (: 1/ (nL–Nl) :). The isotomic of the first point. Of course this is the famous "fourth intersection" of the conics, the other three not being defined in this form of writing circumconics. This shows that this intersection can be constructed as the isotomic of intersection of the two lines in 1.

3. Inconics √(x/l)+√(y/n)+√(z/n) = o and √(x/L)+√(y/N) +√(z/N) = o intersect at

(: (1/√(nL) –1/√(Nl))² :).

By taking all possible signs for the √'s, one gets the four intersections.

I present these in parallel fashion because the mathematics is very much the same. These are or are related to "cross points" if you prefer non-geometric names.

The four intersections of 2 inconic form

one central point

( (1/√(nM) – 1/√(Nm))² : (1/√(nL) – 1/√(Nl))² : (1/√ (mL) – 1/√(Nm))² )

three points that are central as a group and in perspective to ABC at

( (1/√(nM)+1/√(Nm))² : (1/√(nL)+1/√(Nl))² : (1/√ (mL)+1/√(Nm))² ).

There is always a desmic system present. (Pictures of this desmic system can be seen below and at the inconic hub on my webpages. The splash picture for "the isotomic Feuerbach inconics" is such a picture.

Figure: The desmic system formed by the intersections of 2 inconics. The four intersections are green; their mates are red. These 8 points from a projective cube with ABC and a fourth point (blue).

Examples. The Steiner inellipse (perspector G) and the Brocard inellipse (perspector K) have central intersection : b²(c-a)² : and perspector : b²(c+a)²:

Examples. The incircle (perspector the Gergonne point G_o) and the Mandart inellipse (perspector the Nagel point N_o) have central intersection F_o = : (c–a)²s_b :, the Feuerbach point and perspector X55_o = : b² s_b:, a center of similarity of the circumcenter and incircle.

Wilson's treatment uses the conjugation defined by the two points, his formulas using the square root of the invariant point of the conjugation.

Intersections of inconics give natural examples of points defined as square roots of functions of the triangles edges.