Given a general ellipse (with axes not necessarily parallel to the x- or y-axes), is there a compass-and-straightedge method for constructing the major and minor axes?
Just to clarify the question: if you know the equation of an ellipse -- say, something like $4x^2 + 9y^2 - 10xy = 40$ -- it is of course possible using algebra and trigonometry to calculate the angle of rotation of the ellipse's axes relative to the x- and y-axes. But if you instead have only a graph of the ellipse (with, let's say, the center identified), rather than its equation, is there a purely geometric way to construct the axes?
It occurs to me as I write this that if you knew where the foci were, the problem would be trivial. So I suppose an equivalent formulation of the problem would be: Given an ellipse, is there a compass-and-straightedge method for constructing its foci?