OscillatingCircle

Conic main chords and oscillating circle

I call main chord of a conic a chord parallel to an axis of the conic. The following properties are easily obtained by applying the last remark in ReflexionOnConics.html . Let A₁A₂ be parallel to an axes of the conic.
[1] For D on the tangent at A₁, secant DB₁B₂ parallel to the other tangent (at A₂) the chord A₁A₂ is bisector of the angle B₁A₁B₂.
[2] Triangles DA₁B₁ and DA₁B₂ are similar.
[3] DA₁² = DB₁*DB₂ (B₁, B₂ are inverse with respect to the circle (D,|DA₁|).
[4] The circle through A₁, B₁, B₂ is orthogonal to the circle (D, |DA₁|), hence tangent to the ellipse.
[5] Read inversely: Every circle tangent to the ellipse at A₁ intersects it at B₁, B₂ such that B₁B₂ is parallel to the other tangent and A₁A₂ is parallel to an axis and bisects the angle(B₁A₁B₂).

To prove [2] notice that (angle(B₁CA₁) =angle(CA₁B₂)+angle(CB₂A₁) = angle(DA₁C) = angle(DA₁B₁)+angle(B₁A₁C), but angle(B₁A₁C) = angle(CA₁B₂) etc.. The other statements follow from the aforementioned reference.
Notice that circle k, passing through {A₁,B₁,B₂}, for B₁ tending to A₁, tends to the tangent oscillating circle (o) of the conic at point A₁. B₁B₂ tends to the reflected of A₁D, which intersects the conic at a second point A₃.
This remark gives an easy procedure to construct the oscillating circle of a conic at a point A₁:
[1] Draw a main chord A₁A₂.
[2] Reflect the tangent A₁D on this chord to obtain A₃.
[3] The center of the osculating circle (o) is the intersection point of the normal at A₁ and the medial line of A₁A₃.
Notice that, in case there are two main chords through A₁ it doesn't matter which one you select, the above procedure giving in both cases the same point A₃.

Conic main chords and oscillating circle

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