Is it possible to construct the asymptotes of a hyperbola from the points on the hyperbola by compass and straightedge alone? And if so, how to construct them?
I have no idea how to approach the first question. It seems it should be possible as it is similar to constructing a tangent to an ellipse, but I haven't been able to adjust such a construction to work for the asymptotes of a hyperbola. An illustration or a reference would be welcome.
See in https://fr.answers.yahoo.com/question/index?qid=20130810183100AArTMlg Warning, not the first solution but the second one, by Pope, which constructs the hyperbola's centre first, which looks compulsory. For fully understanding it, you need to have seen a minimum of properties of what is called a diameter in a conic curve, whose definition I recall:
When a line is moved paralelly to itsef, i.e., keeping the same direction (D), the midpoints of its points of intersection with the conic section belong to a same line (D') ; if the conic section possesses a centre (this is the case here, or for an ellipse, but not for a parabola) the centre belongs to this diameter. The line with direction (D) passing through the centre is called the conjugate diameter of (D').