Here is my next puzzle! Sorry that it’s been greatly delayed.

I mentioned previously the problem of constructing the midpoint of a given circle, using only ruler and compass. One student solved it nearly immediately (congratulations!), so here I offer a more general problem.

#### Sylvy’s puzzle #5

Given a parabola in the plane, construct its vertex using straightedge-and-compass construction (sometimes called `ruler-and-compass’ construction).

**Straightedge-and-compass constructions**

I want to give a rough description of what it means to perform a `straightedge-and-compass’ construction. We work in the Euclidean plane . You may be familiar with using a ruler and pair of compasses to construct geometrical drawings. For example, if you are given two points you can use the compasses to draw a circle with centre and radius .

Further, you can then construct a regular hexagon inside that circle such that the edge-length of the hexagon is equal to the radius of the circle. If you’ve not done it before, try it! It’s quite fun!

The issue arises: what shapes can one draw in this way? And which points in the plane can arise as intersections of circles/lines that have been constructed in this way?

We start with two points , as above. What can we do with ? We can draw a unique straight line which passes through , we can draw a circle with centre and radius , and we can draw a circle with centre and radius .

These three `circle-lines’ intersect in various points: draw them! The two circles intersect in two points; the line intersects in two points, one of which is ; and the line intersects in two points, one of which is .

Anyway, we get four new points. Using these new points, together with in various ways, we can construct other circle-lines. Then we can consider the intersections of *those*. And so on. All of these points that we obtain are said to be *constructible*. The key point is that there must be some *finite* sequence of points and circle-line constructions to get from to .

For simplicity (and so that we don’t have to worry about our choice of and ), we usually set to be the origin and set to be .

**Deadline: **11am Thursday 3rd of December

(you can either send your solution by email or put a hand-written solution into the folder on my office door)

**Prize: **something chocolatey? I’m open to suggestions…

**Solution: **I’ll post the solution soon after the deadline.

The webpage for these puzzles is http://anscombe.sdf.org/puzzle.html

Have fun!