RectHypeParaChords

Rectangular hyperbola intersections with circles

Consider a rectangular hyperbola and a chord AB of it. For every circle (c) passing through {A,B} the other two intersection points {C,D} of the hyperbola with the circle define a chord that is parallel to a fixed line L₁ and has its middles G on another fixed line L₂.

The existence of L₂ is a consequence of the existence of L₁, since then, by the general theory of conics, the middles of parallel chords are on a line (the conjugate direction to the one of the parallels).
The proof that DC are parallel to a fixed direction follows by a remark made in OrthoRectangular.html . There we saw that considering the triangle ABC, having its vertices on the rectangular hyperbola, the fourth intersection point of the circumcircle with the hyperbola is the symmetric to the center of the hyperbola of the orthocenter H of ABC, which lies also on the hyperbola. Thus, joining the middles, we see that GE is parallel to CH, hence orthogonal to AB hence is a fixed line through E (this is L₂).
In the aforementioned reference we noticed also that the angle of two chords of the rectangular hyperbola equals the angle of the lines joining their middles. This implies also that the angle of lines {AB,DC} is fixed, hence the parallelity of DC to a fixed direction (the conjugate of EG).

Remark The property is valid for any conic. This is proved, using some calculation in coordinates, in the file PowerGeneral.html (last proposition).

Rectangular hyperbola intersections with circles

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