Hyperbola_From

Hyperbola from middles

Consider a hyperbola c and its auxiliary circle c'. Draw two tangents {t, t'} parallel to asymptotes and
intersecting at a point P. Then every tangent p to the auxiliary circle intersects lines {t, t'} correspondingly at
points {Q,R} and the conic at two points {S,T} of which one, S say, is the middle of QR.

The proof starts by defining the polar FE of P with respect to the conic c. This is simultaneously the polar of P with respect to
circle c' since (G,H,I,P) = -1 are harmonic with respect to either of the curves. Take then a tangent of c' at B as required and
consider the intersection point A of CB with the polar EF. The polar p_A of A with respect to the circle passes through P (by the
reciprocity of relation pole-polar) and is parallel to tangent p. Besides (E,F,J,A) = -1 build a harmonic division, thus the bundle of
lines at P: P(E,F,J,A) defines a harmonic division on every line it meets. Apply this to the tangent p. Since PJ is parallel to this
tangent, S is the harmonic conjugate with respect to {R,Q} of the point at infinity of line p, hence it is the middle of RQ.

Remark This property has an inverse characterizing a certain geometric locus: Given a circle c' and two tangents {t,t'}, the middles
S of segments RQ intercepted by tangents p of the circle on {t,t'} generate a hyperbola c having c' as auxiliary circle and asymptotes parallel to
the given lines {t,t'}.

The proof of this inverse can be easily supplied by constructing the hyperbola from the given data and then using its proven property
identify it with the required locus. A similar property defining hyperbolas is discussed in HyperbolaProperty.html .

Hyperbola from middles

See Also