Pencil of conic sections

Given points A, B, C, to study the pencil of conic sections tangent in B to AB and in C to AC. Locus of center and focii.

The locus of centers is the median from A in triangle ABC (a simple question of pole/polar : the median is the polar of the point at infinity on line BC, hence contains the center = pole of line at infinity)
In the following applet, the choice of the conic section in the pencil is therefore driven by the draggable point O, center of the conic section, on the median OM.
Then through a classical construction, the axes are constructed and then the focii.
The purpose of this applet is to illustrate the pencil of conic sections of type III (tangent in B and C to fixed lines).
When O is in G, it is the inner Steiner ellipse in triangle homothetic of ABC in ratio 2 (light green).

We then draw the locus of the Focii, which seems to be a strophoidal curve (grey).
The holes in this locus is because of the limited range of O on the line OG, and of some loss of accuracy (not enough points on the locus).
Another hole appears when the conic section is a rectangular hyperbola (due to inaccuracy in the choice of axes)
Spurious "spikes" or holes also appear when the conic section degenerates (when O on line BC or near A), or when the strophoid itself degenerates into a circle + line etc...

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Details about the construction :

The conic section is drawn using the obvious additional points D and E symmetric of B and C through chosen center O
This defines the conic section through 4 points and a tangent, and the usual construction using Pascal theorem is used :
A current line Bt is freely chosen (drag point t) and intersects CD in I
Lines BC and DE are parallel, that is intersect at a point J at infinity.
Line IJ is therefore the parallel to BC in I.
It intersects line AC (the tangent in C) at K
EK and Bt intersect at the current point P of the conic section.
Pascal theorem says that the pairs of opposite sides in hexagon BCCDEP intersect in I,J,K aligned.
C is counted twice and "line CC" is the tangent in C.

Two pairs of conjugate diameters are obtained :
the first pair is the median AO, and the parallel in O to BC
An other pair is constructed, the point P on the conic section on a parallel to CO through B is constructed as a specific current point (as above, with Bt // CO)
Let M the midpoint of BP. The pair of conjugate diameters is (OC, OM)

The two pairs of conjugate diameters are projected on any circle going through O, in (x1,y1) and (x2,y2)
For ease of construction the "any circle" is centered in C (why not)
The intersection S of x1y1 and x2y2 is then the pole of the involution xi↔yi on the circle.
Diameter CS of this circle intersects it at u and v, and Ou, Ov is a pair of orthogonal conjugate diameters, hence they are the axes.

Now we get the focii as follows :
The tangent AC and the perpendicular to AC in C (normal in C) intersect the focal axis in I and J.
(F,F',I,J) is an harmonic range, that is OF² = OF'² = OI.OJ
So that OF = OF' is the length of the tangent from O to the circle with diameter IJ.
The focal axis is that one of Ou, Ov for which I and J are on the same side from O

The projection H of the focus F on the tangent AC lies on the principal circle, which gives the focal vertices.
The perpendicular to focal axis in F intersects the principal circle in Z, giving the transverse vertices.
Excentricity is then OF/Oa

The locus of F is drawn by sweeping O on the median line.
When triangle ABC is isosceles, this locus degenerates into the circumcircle of ABC + the median.
When O goes to infinity, the conic section is a parabola. Its focus (in grey) is directly obtained as intersections of "light rays" parallel to the median and reflected on AB and AC at B and C.
To get clean drawings of the conic sections and the locus of F,F', we should sweep O separetly in different regions of the median, carefully avoiding the neighbourhood of A and of the midpoint of BC.
Also the "any circle" used to construct fails when one of u or v becomes very near of O : the axis Ou or Ov is then undefined.
About spikes in drawing the conic section itself (especially hyperbolas) we should also sweep the line Bt used to draw the conic section in separate ranges (to avoid points "at infinity")

Other cases

We may wonder of the locus of focii for other pencils of conic sections
The locus of the center is well known to be a conic section, and no more a straight line.
Therefore for generating the pencil, instead of chosing the center, we just chose any "5th" point (on a given line for instance)
The following cases show the locus of focii in case of
  1. Type I through 4 (real) points
  2. Type II through 3 points and a tangent in one of them

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    The pencil is defined by the points A,B,C and thge tangent in A
    The conic section in the pencil is chosent through point D on the normal in A.

  3. Type III is already studied : 2 points and their tangent
  4. Type IV is 1 point and an osculating tangent at a second point
  5. Type V is 1 point and a "surosculating" tangent at this point
to be completed...

 

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