ElevenPointsConic

1. Eleven points conic

Given two angles CAD, EBF at points A and B and a line (e). There is a remarkable conic passing through eleven points related to these lines. The conic is created as the locus of intersection points W of lines AW, BW which are defined as the polars of a variable point V on (e) with respect to the angles CAD and EBF respectively.

The locus of points W contains obviously the following 5 points. {A,B,Q,P,R}. Point Q is the intersection of lines IG and HJ and {P,R} are respectively the harmonic conjugates of M w.r. to {I,G} and U w.r. to {H,J}. As seen in the figure GHIJ is the quadrilateral formed by the two angles.
That this locus is a conic follows easily by a simple computation in projective coordinates, for example in the basis {A,B,Q,R}.
It is even not necessary to carry out the computations. It suffices to notice that the coordinates of W satisfy a quadratic equation, hence W lies on a conic. This conic then is the one passing through the above five points.
It is also readily seen that the harmonic conjugates T=E(H,I), S=D(I,J), N=F(G,J), O=C(H,G) are on this conic.
Finally the common conjugates {K,L} to {E,F} and {C,D} are also on the conic.

2. Related family of conics

To the above configuration relates the family of conics (d) passing through the four intersection points {G,H,I,J} of the sides of the two angles.
It is namely readily seen that the poles of line (e) with respect to all these conics satisfy the condition of definition of conic (c), hence they are points of this conic.
Another point of view, related to the fact that (c) is a circumconic of the trianlge ABQ is discussed in ElevenPointConic.html .

3. Nine points conic

The previous picture delivers an interesting particular case, when line (e) coincides with the line at infinity. Then point V defines parallels from points A, B and W is the intersection point of the conjugates AW, BW to these parallels with respect to the two angles.
In this case points {N,O,T,S,P,R} are middles of respective sides {GJ, GH,HI, IJ, IG, HJ}.
In this case also the poles of line (e) with respect to the conics through {G,H,I,J} are the centers of these conics. Thus (c) becomes the locus of centers of conics passing through these four points.
Of particular interst are the double points K, L in this case. If existent they imply that (c) is a hyperbola and they determine the asymptotic directions of this conic. This point of view is also discussed in NinePointsConic.html .

4. Desargues involution

Points K, L are also the fixed points of the Desargues involution defined by the four points {G,H,I,J} on line (e). They determine the contact points of the conics passing through these four points and tangent to line (e). This subject is discussed in DesarguesInvolution2.html and also in FourPtsAndTangent.html .

References

Joachimsthal, F. Elemente der analytischen Geometrie. Berlin, Reimer, 1871, p.176.

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1. Eleven points conic

2. Related family of conics

3. Nine points conic

4. Desargues involution

See Also

References