# Puzzle corner: Sylvy’s puzzle #5

Cross-reference to puzzle page on my website

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 $\mathbb{R}^{2}=\mathbb{R}\times\mathbb{R}=\{(x,y)\;|\;x,y\in\mathbb{R}\}$. You may be familiar with using a ruler and pair of compasses to construct geometrical drawings. For example, if you are given two points $A,B$ you can use the compasses to draw a circle with centre $A$ and radius $|AB|$.

Further, you can then construct a regular hexagon inside that circle such that the edge-length of the hexagon is equal to the radius $|AB|$ 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 $A,B$, as above. What can we do with $A,B$? We can draw a unique straight line $L$ which passes through $A,B$, we can draw a circle $C_{A}$ with centre $A$ and radius $|AB|$, and we can draw a circle $C_{B}$ with centre $B$ and radius $|AB|$.

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

Anyway, we get four new points. Using these new points, together with $A,B$ 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 $A,B$ to $C$.

For simplicity (and so that we don’t have to worry about our choice of $A$ and $B$), we usually set $A$ to be the origin $(0,0)$ and set $B$ to be $(1,0)$.

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!