Graphically locate the axes or foci of an ellipse from 5 arbitrary points on its perimeter.

Solution 1:

Given five points of the conic, and using Pascal's theorem you are able to find the intersection of the conic with an arbitrary line that contains one of these. Moreover, you are able to find the tangent to the conic at any of these points.

Now, you are able to produce pairs of conjugate diameters that intersect at the center of the conic (ellipse). To do that, notice that the middle of the chords parallel to a given direction are all on the line passing through the center of the ellipse (the conjugate diameter). This produces conjugate directions, to get the sizes of the diameters one uses Prop XXII from the source below. Up to now, we haven't mentioned angles.

The rest, and most interesting part is explained in an old book that you can find here: An elementary treatise on the Geometry of Conics, page 96, ex 8. There îs one beautiful basic fact being used , that the product of the intercepts on the tangent to an ellipse by a pair of conjugate directions is constant (equals the square of the half the diameter parallel to the tangent). With this and some classical construction one is able (ex 8 mentioned above) to construct a pair of conjugate directions that are perpendicular, so the axes. One now uses a consequence of prop XXII in the same book to get the sizes of the diameters.

I noticed now that one can also use results from page 106 of the same book (ex 14 or 15).

Also see Rytz's construction.

All these results seem classical, to be known by Newton and maybe even before him (Apollonius?), but forgotten these days, since the geometry of conics is nowadays seen as a particular case of "analytic geometry", excepts perhaps the "fun things" like Pascal, Brianchon, Poncelet and very basic things.

Solution 2:

Given just 5 points that are known to lie on the perimeter of an ellipse, and generating that ellipse using the method described in the partial answer offered here by @Ng, and then continuing with the instructions found at http://whistleralley.com/conics/conic_construction/ellipse_parts/, it is possible to construct the entire ellipse, its major and minor axes, center, vertices, foci, and directrices. Therefore, this is a valid solution to my question as posed, however the initial construction of the ellipse still requires using more than just a compass and straight edge, so I am still hoping for a ruler-and-compass only solution. Thanks again @Ng for supplying the first steps that I was missing.