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.

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 .

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 .

ElevenPointConic.html

FourPtsAndTangent.html

NinePointsConic.html

ProjectiveBase.html

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