From 1 Focus to conic

Suppose I gave you one point related to an ellipse, say its center, and asked you to draw the ellipse. You might try very hard, but I have not given you enough information. But if I gave you that same information for a triangle conic, say one inscribed in the triangle, the conic is completely determined. If the center is in a particular spot, the axes must point a certain way; there is only one possibililty. The same is true if I gave you, say, only one focus.

The situation is somewhat different for circumconics. They are completely determined by knowledge of their center, but knowing one focus only narrows the possibilites down to four! When I found this out, I had to understand it; here is the results of the Hyacinthos geometry discussion of the matter. The participants were Jean-Pierre Ehrman, Nikolaos Dergiades, Wilson Strothers, Peter Moses, Jim Parish, and myself.

One of the most interesting discussions lately has been about the foci of conics. This discussion is only about conics related to the triangle, especially inscribed and circumscribed ones.

Projective properties of a triangle conic.
Perspector -- this single point is all that is needed to construct an inscribed or circumscribed conic.

Euclidean properties of a triangle conic
Axes, vertices, foci, and directrix.
The properties are more of a problem because the foci are so hard to compute from the

Nikolaos Dergiades began the discussion by pointing out that because of a special property of foci of inconics, they could easly be constructed by knowing a single focus.

Given one focus F, construct the other as its isogonal conjugate gF. The center C of the conic is the midpoint of these two points. From the center, the perspector P = dt C can be found. From the perspector, the conic can be constructed.

What we see by dragging F and the triangle vertices: We look for the places where the conic degerates. It only does this when F crosses an edge. Hence any formula for for the area, which must be symmetric in triangle variables, must have a factor of xyz, the product of the three coordinates which goes to zero when F is on a side. I learned this method from Jean-Pierre Ehrman.

The triangle with one known focus F.
Construct the isogonal conjugate gF
Now the pedal triangle of gF. The pedal triangle are the feet of gF on the sides, created by dropping perpendiculars from gF to the sides. The notation B_[gF] indicates the B vertex of the pedal triangle of gF. The sides of the triangle are lines rather than segments to give a good construction.
Now construct the 4 incenters, also known as the in- and ex-centers, of the gF-pedal triangle. Red lines are the bisectors of the pedal triangel. The notation Io_[gF] indicates the original incentter of the pedal triangle of gF.

The line a' is the perpendicular through A to gF A_Io[gF]. This line is tangent to the required circumconic.
The circumconic is constructed as the inconic of triangle ∆' = A'B'C'. The second focus F_o is the isogonal conjugate of F with respect to ∆'. The center C_ois the midpoint of the two foci. The perspetor P_o = tdC_o
The a-conic The yellow triangle is ∆', the constructed triangle to which the conic is inscribed.
The b-conic
The c-conic

Herre are all 4 conics indexed as o,a,b,c. The points boldly labeled F is a focus of each conic and these four are the only possible ones that have F as focus. Each is colored differently. Each conic meets the each other four times, three at A, B, C and a fourth time.

There are 4 perspectors P_x (x = o, a, b, c), four centers C_x, four second foci F_x, and 6 fourth intersections. These points have some interesting relationships to eachother.

Each conic meets the others at A, B, C and a fourth time, the famous fourth intersection of two circumconics. We label these intersections with the two indices of the conics that form the intersection, as in "ab." Here is a picture of the conics with the labeled intersections.

There are 6 intersections labeled oa, ob, oc, bc, ca, ab. We analyze these points and their relationship to ABC.

There are two groupings symmetric in the triangle: oa, ob, oc and bc, ca, ab. These two triangles are in perspective at a point labeled o.

There are three points on each conic. For example oa, ab, ac are all on the a-conic. This triangle is colored yellow in the picture. The remaining triangle ob, oc, bc whose vertices have no "a", is in perspective with ABC at a point called "b." oa, ab, ac and ob, oc, bc are in perspective at the focus F. There are three of these pairs of triangles, giving 4 new points, labeled o, a, b, c.

The four new points are connected by 6 lines, each containing two of the new points and one vertex.

The four perspectors have a special relationship to ABC. If each vertex is connected to all four points, four new concurrances occur as shown here.

If lines are colored appropriately, we see that we have the 4 vertices of a projective cube.