## 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.