Constructions: it's all about the rhombus

Constructions seems to be an ideal topic in which links can be made with other parts of geometry.  It also seems a good opportunity to teach for understanding (in a Skemp-type way) as well as leaving pupils with a method they can use.

I learned this when I was at school (in the 20th century).  I remembered where to put the point of the compasses and which arcs to draw so I could carry out the constructions accurately, but I think there is more to it than that.

First off: what counts as a 'construction' and what doesn't?

If you are asked to draw an "accurate" version of a triangle when given the three sides, it is usual to use a pair of compasses.  (Draw one side - then make an arc from each end - where the arcs cross is the third vertex.)  This will give a pretty good version of the triangle (assuming the compasses don't slip and you have a sharp pencil), but this won't ever be entirely accurate because measurement is involved.  This isn't usually considered to be a proper construction for this reason.

Euclidean constructions involve using a 'straight edge' (a ruler - but not using it for measuring) and a pair of compasses (for drawing arcs).  [These also won't be exactly accurate because the compasses might slip, there is the thickness of the pencil to consider, etc, but the idea here is that this method will, in an ideal situation, give an idealised answer, so it considered to be a purer construction.]

The main constructions involve bisecting an angle and creating the perpendicular bisector to a line segment.  Arguably, all of the others build on these.

The big idea here is that they both involve constructing a rhombus.

There are three properties of a rhombus that we need:  

1. A diagonal of a rhombus bisects the angle
2. The two diagonals of a rhombus bisect each other
3. The diagonals of a rhombus are perpendicular

Consider how to construct an angle bisector.  Here is an angle:

When we draw in the arcs using a pair of compasses then we are actually constructing a rhombus:

We know this is a rhombus because we have set the compasses to a particular length, and then each of the sides of the rhombus is the same as that length.

Because of the properties of a rhombus, if we join the opposite vertices we bisect the angle.

Now consider how to draw the perpendicular bisector of a line segment:

Again, we need to draw arcs and construct a relevant rhombus:

The original line segment becomes one diagonal of the rhombus, with the line we will draw in becoming the other diagonal.  The properties of a rhombus tell us that the diagonals bisect each other at right angles, so we have got the perpendicular bisector.

There are lots of other constructions, such as constructing the perpendicular to a line that passes through a particular point, but this can be done by putting the compass point on the point and drawing an arc that intersects the line twice.  Now use the line segment between the places where the arcs cross the line and use the second construction above.

One way to teach this is to start by getting pupils to construct lots of rhombi.  Then they can sketch the rhombus they need onto the diagram, and finally they can construct the relevant rhombus.

Remember: it's all about the rhombus!