quality:correction
Midpoints of parallel chords

The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram).

The points of any chord may lie on different branches of the hyperbola.

The proof of the property on midpoints is best done for the hyperbola y = 1 / x {\displaystyle y=1/x}. Because any hyperbola is an affine image of the hyperbola y = 1 / x {\displaystyle y=1/x} (see section below) and an affine transformation preserves parallelism and midpoints of line segments, the property is true for all hyperbolas: For two points P = (x 1, 1 x 1), Q = (x 2, 1 x 2) {\displaystyle P=\left(x_{1},{\tfrac {1}{x_{1}}}\right),\ Q=\left(x_{2},{\tfrac {1}{x_{2}}}\right)} of the hyperbola y = 1 / x {\displaystyle y=1/x}

For parallel chords the slope is constant and the midpoints of the parallel chords lie on the line y = 1 x 1 x 2 x. {\displaystyle y={\tfrac {1}{x_{1}x_{2}}}\;x\.}

Consequence: for any pair of points P, Q {\displaystyle P,Q} of a chord there exists a skew reflection with an axis (set of fixed points) passing through the center of the hyperbola, which exchanges the points P, Q {\displaystyle P,Q} and leaves the hyperbola (as a whole) fixed. A skew reflection is a generalization of an ordinary reflection across a line m {\displaystyle m}, where all point-image pairs are on a line perpendicular to m {\displaystyle m}.

Because a skew reflection leaves the hyperbola fixed, the pair of asymptotes is fixed, too. Hence the midpoint M {\displaystyle M} of a chord P Q {\displaystyle PQ} divides the related line segment P ¯ Q ¯ {\displaystyle {\overline {P}}\,{\overline {Q}}} between the asymptotes into halves, too. This means that | P P ¯ | = | Q Q ¯ | {\displaystyle |P{\overline {P}}|=|Q{\overline {Q}}|}. This property can be used for the construction of further points Q {\displaystyle Q} of the hyperbola if a point P {\displaystyle P} and the asymptotes are given.

If the chord degenerates into a tangent, then the touching point divides the line segment between the asymptotes in two halves.