Conic Challenges

These are some geometry construction challenges relating to conic sections. This began with a single document in The Geometer’s Sketchpad. Though no longer marketed as it once was, Sketchpad still has a team of developers hard at work. The latest version can now be found at The Geometer’s Sketchpad. There may also be other dynamic geometry applications that can be used for this, but I am not so familiar with them.

Each page of the document includes certain given objects, and the challenge is to use those objects for completing the construction of some certain conic section. Ideally the constructed figure should stay intact when the given objects are dragged into other positions (the drag test). It should vanish entirely in those cases where no solution is possible. Also, I would have all solutions include the center, foci, vertices, axes, directrices, and asymptotes, as they apply. Solutions to the challenges will be ellipses, hyperbolas, or parabolas. Degenerate cases are not addressed. A circle may be considered a conic section, as a special case of an ellipse, but circles are not addressed either. Strictly speaking, none of these actual curves can be constructed by the classic compass-and-straightedge standard, but all of these associated points and lines can be constructed, and the dynamic geometry will take it from there.

The study of conic sections has lately become somewhat marginalized in modern education. If they are presented at all, they tend to appear in a rather simplistic, analytic form. The parabola, for example, is shown only as the graph of a quadratic function. Many students (and even teachers) have come to believe that all conic axes are parallel with one of the coordinate axes. Coordinates do not appear here at all, and analytic solutions are discouraged. The curve may have any orientation, so long as it stays in the plane. I tend to avoid horizontal and vertical orientations in order to emphasize that point.

I tend to make frequent use of the term “axial vertex”, which most geometers would simply call “vertex”. Thank Apollonius for that. In his Conica any point on the section might be called a vertex, and may define a unique diameter and ordinate direction, while a vertex on an axis is a special case. I cite Conica so often, it only seems right that I sould use his terminology.

Here is the Sketchpad document carrying the challenges, 26 in all. Solutions are below, but try them yourself first.

Conic Challenges.gsp

The conic challenges are separated into four groups, as are the solutions, which may be reached with the links below.

Parabola
Ellipse
Hyperbola
General

The solutions in these pages are mostly only outlines. A certain competency in geometric construction is assumed. Still, here are a couple that did not appear in my ninth-grade geometry course, but come up often here. They might save someone some time here.

Geometric Mean

Let point A lie between B and C, to construct the geometric mean of AB and AC, construct the circle with diameter BC. Construct D on that circle, such that AD is perpendicular to BC.

Thus, AD is the geometric mean of AB and AC.

Harmonic Division

Let point C lie on line AB. To construct point D such that CD divides AB harmonically, begin with the circle having diameter AB. Let the perpendicular bisector of AB meet the circle at E. Construct lines EB and EC. Reflect EC across EB, and let the reflection image meet AB at point D.

It can now be said that AB and DC divide each other harmonically. There is no harmonic division if C is the midpoint of AB, but this construction holds up for any other point on AB, inside or outside the circle. There are many other constructions that would do as well, but it is hard to beat this one for simplicity.

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