is a question-and-answer website that focusses on substantive discussion of mathematical questions, with participants ranging from students to teachers to professional mathematicians. The question What is the (mathematical) point of straightedge and compass constructions? was asked in 2011 and has accumulated a number of interesting responses (representing a wide range of viewpoints). One of the answers argues:

The beauty of straightedge and compass constructions, as opposed to the use of, say, a protractor, is that you don’t measure anything. With ruler and compass you can bisect an angle without knowing its size, whereas with a protractor, you would have to measure the angle and then calculate the result.

In other words, the point of this form of geometry is that it can be done independently of calculations and numbers. I think this is an important idea to teach: mathematics is not about numbers, but about objects adhering to certain rules (axioms).

Math.SE user Greg Graviton

What’s the game?

We start by describing how the ruler and compass are intended to be used.

Ruler and Compass Construction Rules

  1. Given any two points $A$ and $B$, the ruler can be used to draw the line $L_{A B}$.
  2. Given a point $O$ and a length $r (r>0)$, the compass can be used to draw a circle with center $O$ and radius $r$.

Notice that (1) corresponds to Euclid’s first axiom of geometry (through any two points there is a line). Also, the “ruler” in (1) is just a straight-edge – we do not use markings on the ruler to measure distances. Number (2) corresponds to Euclid’s third axiom (existence of circles with a given radius and center). In practice, we often don’t use the compass to draw a complete circle – instead, we just draw one or more parts of it (arcs).

In another response to the question raised above, user Joseph Malkevitch observes:

Throughout the history of mathematics there has been a tension between the issue of showing that something exists and actually constructing what one may know is there.

Math.SE user Joseph Malkevitch

Hopefully the correspondence between the construction rules and the axioms of geometry give some hint at the relationship between geometric constructions and proofs — if we want to prove that certain geometric object exists (for example, a perpendicular bisector of a segment), then we can simply provide a ruler-and-compass construction to build that object.

Give it a try

Use a ruler and compass to complete the following constructions.

1. Copy a segment

  1. Draw three points, A, B and C anywhere on the page.
  2. Draw the line segment $\overline{AB}$.
  3. Construct a copy of $\overline{AB}$ starting at point C.

2. Copy an angle

  1. Draw two rays $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ with a common endpoint P to make an angle.
  2. Make a copy of the angle at another place on the page. Start by drawing another ray $\overrightarrow{ST}$.
  3. Now construct the ray $\overrightarrow{SU}$ so that $\angle QPR \cong \angle TSU$

3. Construct an equilateral triangle

  1. Draw two points A and B.
  2. Construct $\triangle ABC$ with $\overbar{AB} \cong \overbar{BC} \cong \overbar{CA}$.

4. Midpoint and perpendicular bisector

  1. Draw two points P and Q and the line segment $\overbar{PQ}$.
  2. Construct a line L that is perpendicular to $\overbar{PQ}$ and passes through the midpoint of $\overbar{PQ}$

5. Bisect an angle

  1. Draw two rays $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ with a common endpoint P to make an angle.
  2. Construct another ray $\overrightarrow{PS}$ so that $\angle QPS \cong \angle SPR$

More Construction Challenges:

Construct a scale drawing of a triangle with a scale factor of r=2.

Given a line L and a point P, construct a line parallel to L passing through P.

Find the center of a rotation given a segment and its image under the rotation.