Lemma 1
Suppose that H is a hyperbola with centre C.
Let U be a point not on H, other than C
Let the parallels to the asymptotes through U meet H at V,W.
Then
(1) the mid-point of VW lies on UC, and
(2) the polar of U is parallel to VW.
Proof
Let the infinite points on H be I,J, and let V lie on UI, W on UJ.
Now consider the quadrilateral VIJW.
By definition, VI, WJ meet in U.
As I,J infinite, IJ, VW meet at K, the infinite point of VW.
Say VJ, WI meet at X.
Then, by a standard result, the polar of any of U,X,K is the line
joining the other two.
As K is infinite, the polar UX passes through C.
Now, as I, J infinite, VUWX is a rectangle, so the mid-point of VW
is on UX, ie on UC.
Also the polar of U is XK. This has infinite point K, as has VW.
We note that, in the notation of the proof, X is the reflection of U in VW.
Lemma 2
Suppose that V,W are defined from U as in Lemma 1,
and that V',W' are defined from U'.
Then UVW and U'W'V' are in perspective.
Proof
We use the notationof Lemma 1.
Consider the hexagon VIV'WJW'.
VI and WJ meet at U,
IV' and JW' meet at U',
Let V'W and W'V meet at Z.
By Pascal's Theorem, U,U',Z are collinear.
From the definition of Z, this says that UU', V'W, W'V concur at Z.