Constructions!

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(I had a colleague --in Communications-- who objected to an exclamation point used in a poster for a mathematics colloquium.  He was of the opinion that exclamation points had absolutely no place in mathematics, and constituted a violation of the principle of truth in advertising!)

Anyway, constructions are a truly fun part of traditional geometry.  The idea is: how to build various geometric figures using only straight-lines and circles, or "ruler and compass," to quote the traditional phrase.

Here is a simple introduction to the subject.  It has the form of a game, with very stringent rules:

1.  Given a line segment, you can make a circle with its center at either end, passing through the other.  (In other words, your segment becomes one of the radii of the circle.)

2.  Given a line or circle, you can select any number of points on them.  (Actually, you can select any point that has not already been selected.)

3.  You can join any two points with a segment, a ray, or a whole line.


Well, that's it!  Obviously, you can also use a construction that you've described earlier, as part of a later construction.

Well, just to show you how this works, here's a simple example.

Midpoints
Given a line segment AB, it would seem that it is a simple matter to find its midpoint.  But the question is: can it be done using only rules 1, 2 and 3?  The diagram should make the following clearer.

Given:  The segment AB.

Step[1] Make a circle with center at A, going through B.

Step[2] Make a circle with center B, going through A.

Step[3] Name the points where the two circles cross C and D. 
(Does not matter which is which.)

Step[4] Join CD with a line.

Step[5] Name the point M where CD crosses AB. 
This is the required midpoint!

(Please excuse the gratuitous use of the exclamation mark above.  I shall feel free to use them at my discretion hereinafter.)

Once you get the hang of this and you realize that all that is happening is circles and lines, a simple diagram --or a sequence of diagrams-- is all you need to follow the construction.  Just click on the diagram (or double-click), and see whether the construction is clear.

The next two constructions: finding the bisector of an angle, and constructing a perpendicular both use this one.

Note: the constructions have very important geometric implications, but I'm not going to emphasize those.  I want it to be clear that this is just for fun, and there is no hidden educational agenda here, at least at present.

A.