I am studying the (somewhat old-fashioned) "Analytical Conics" by D.M.Y. Sommerville (3rd edition, 1933). (It's fascinating, but the order of presentation of material makes it come across as muddled sometimes).
I have found this definition in Chapter V: The Hyperbola, section 2.
"… chords through the center are diameters."
Okay, straightforward enough: he does not actually define "chord", but by extrapolating from its definition for the circle I assume he means a line segment joining 2 points, whether they be on the same branch or not. Hence this definition is simple and straightforward.
However, when I seek to confirm this online, I get the following mumbo-jumbo, usually accompanied by a diagram in which the line so constructed is anything but the locus of midpoints:
"The locus of the mid point of a system of parallel chords of a hyperbola is called a diameter of the hyperbola and the point where the diameter intersects the hyperbola is called the vertex of the diameter."
Now I have run it up on GeoGebra, constructed a system of parallel chords, constructed a line through its midpoints, and indeed, it passes neatly through the centre:
… and indeed, it is definitely the case that the locus of the midpoints do in fact form a diameter.
The question I have is: if it is indeed blindingly simple to define a diameter as (the infinite production in both directions of) a chord which passes through the centre, why use the complicated definition?
And, having used this definition, why accompany it with a horribly misleading diagram? Examples of such are ubiquitous:
Is there a subtlety here by which it is not always the case that "a chord through the centre (produced appropriately) is the locus of a system of midpoints of parallel chords"?
In due course I will work out the algebra to prove the above, and satisfy myself that it is indeed true (I can't find it in Sommerville — but then its presentational style is dense and I haven't studied the chapter on hyperbolas line by line yet). I will no doubt also find it is true for all conic sections, as I believe it is a projective property: it's obviously true for circles, and I expect there is a form for the parabola that allows it to make sense for it there as well.
Now I understand, from the language in which many of such pages are couched, that this is something which is often pitched at elementary-school level or perhaps high-school level. Here, for example, is a particularly patronising example:
https://www.toppr.com/ask/content/story/amp/diameter-of-hyperbola-52033/
… so it is obviously something which is considered pretty basic, mathematically speaking (although I can't remember anything about this in my own long-ago formal studies of geometry via the Kleinian approach).
Is anyone able to shed any light on this from the point of view of a coherent approach? That is, the equivalence of the definitions of a diameter being straightforward to demonstrate and (presumably) fairly straightforward to prove (although I haven't done this formally myself to my own satisfaction), why is it (in general) not even mentioned in the literature?
EDIT: I have checked for a parabola, and it appears (as you'd expect by a projective argument) that the "diameters" of a parabola (as defined by the locus of the midpoints of parallel chords) are in fact parallel to the axis. This is consistent with the "centre" of a parabola being the point at infinity of the axis of the parabola.
Best Answer
I've spent some time over the last few days working on this.
There is good reason to define a diameter as the locus of the midpoints of parallel chords.
This is because the definition as "chord which passes through the center" is inadequate as a definition for the following reasons:
a) It does not encompass the case of the parabola, where it is straightforward to demonstrate that the locus of the midpoints of parallel chords is a straight line parallel to the axis of the parabola. Thus, and because there is no center of a parabola (unless you go down the rocky road of defining the center as the point at infinity, and there is no real need to do that here) it is meaningless (or at best cumbersome and complicated) to define a diameter in relation to such a point.
b) In the case of the hyperbola, it does not take account of the diameters which are the loci of parallel chords either ends of which are on opposite branches of the hyperbola.
Hence the "chord through center" definition misses these important diameters which do not intersect the hyperbola at all.
So the question is resolved. The reason for defining a diameter as the locus of the midpoints of a system of parallel chords is because it is universally applicable.
The fact that a diameter passes through the center of the (central) conic section is a direct consequence of the above definition.
It would be so nice, though, if more texts (and websites) took the trouble to explain the above.
It would also be nice if the websites would construct adequate and accurate diagrams illustrating the concept without forcing the readers to scratch their heads and say: "But those are not the midpoints ..."