I have written about geometric construction a number of times on this blog, and on my website. I outlined my philosophical outlook on this topic here.
Today I summarize some construction activities, a subset of the unit you can find on my website.
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Geometric construction with straightedge and compass is a standard topic in high school geometry classes. Typically, students are shown specific procedures to construct bisectors, right angles, and so on. Often, they are encouraged to create aesthetically pleasing designs as a follow-up application. There is nothing wrong with that, but it barely scratches the surface of what can be accomplished. Perhaps reading this post will encourage you to go deeper.
After a basic introduction to a few classic constructions, and the proof that they work, students can be asked to solve construction puzzles of increasing difficulty. Besides the fact that this is highly motivational, this departure from a strictly procedural approach helps to strengthen students’ problem-solving ethic and their grasp of standard geometry topics.
The structure of such puzzles is always “Given…, construct…” Access to interactive geometry software is assumed (e.g. GeoGebra). Here is one possible puzzle sequence:
Given two points, construct three different circles that go through both.
Given three points, construct a circle that goes through all three.
Given a line and a point not on it, construct a circle centered at the point and tangent to the line.
Given two lines, construct three different circles tangent to both.
Given a triangle, construct a circle tangent to all three sides.
This set introduces the circumscribed circle, and suggests the need to prove that the perpendicular bisectors do all meet in one point. And likewise for the inscribed circle and the angle bisectors.
There are many more possibilities to address various parts of the curriculum. For example, the logical hierarchy of the named quadrilaterals is mirrored in construction constraints: a correct rectangle construction in GeoGebra can be used to show a square, but a correct construction of a square cannot be used to show a generic rectangle. Likewise for parallelogram and rectangle, and so on.
And here is a challenge that takes the student a little further. The figure shows three circles through point F, tangent to line d.
Two puzzles:
Given F and d, show how to construct those circles.
Construct a fourth circle through F, tangent to d.
As it turns out, constructing a fourth circle needs to rely on a generalizable strategy. (Hint: choose a point T on d, and find a circle tangent to d at T, that passes through F.)
The centers C of the circles students discover in this challenge all lie on a parabola with the line as its directrix and the point as its focus. GeoGebra makes it possible to see what happens to C as you drag T along d by tracing, or by using the Locus tool. ( See Geometry of the Parabola for more on this.)
This falls somewhat beyond the standard geometry curriculum, but it is worthwhile extension: students should know that the parabola is also a geometric object — not just the graph of a quadratic function. The software allows the user who successfully solved this puzzle to create a parabola and see what happens to it when the focus is moved or the directrix is rotated. A further extension would be the search for constructions of the other conic sections.
If you want to find out more about all this, I included a dozen links to relevant posts and pages here.
