Side Story 2-1: Do You Construct, or Do You Freehand?

Newton by William Blake. Public domain.

By using only compass and straightedge, you become part of an ancient tradition.

Quick Question:
Do you prefer using a compass and a straightedge to draw geometric diagrams, or would you rather freehand?

Most of the people I know freehand. I would posit that the simple reason is that geometric construction isn't a topic touched upon at great length in most of our classrooms, and dabbling with the compass is mostly done in drafting class (that is, if you had one; I didn't.). When it comes, however, to people who deal with more elaborate [Difficulty 4+ in this blog] geometry problems on a regular basis, the question does become a tad more relevant. The figures get quite complicated, and for some, the best way to make any sense out of anything is to draw them as precisely as possible. Hence constructing.

I grew up constructing. Michael Serra's Discovering Geometry was probably the text that exposed me, during grade school years when my preferences were most malleable, to the classical art of "geometrizing" using compass and straightedge. (That text also had sections involving the use of translucent patty paper, which I never really developed an appreciation for.) It wouldn't be until late high school until serious construction became a method for a significant number of my co-trainees. By then I had really gotten the hang of it. In a way, it was my specialty; I loved to construct, and I loved geometry problems probably only because I could make them easier by constructing.

My nice little worldview was shattered when I eventually came across people (peers and profs) who advocated freehanding as an actual alternative to constructing. In a nutshell, they viewed compass-and-straightedge construction as training wheels, that relying on them too much meant relinquishing the power of pure imagination in favor of the strictures of graphite and metal. (Interesting callback to the painting above, with "Newton" fixated on his neat diagram, ignorant of the beauty surrounding him. But again I digress.)

Anyway, being years removed from my contest days, I can now provide a (hopefully) sufficient defense for each approach:

Why you should construct

• If two triangles are similar, or three lines concurrent, it should show immediately in a well-drawn figure. Any time you spent in making the figure are more than made up for the time saved in finding the key point.
• Sometimes a guiding line or circle you draw to construct the actual figure really needs to be drawn to solve the problem. It happens very often!
• Don't underestimate the power of a good visual aid. Few things beat a well-drawn diagram in clearing up the problem-solving mind.
• By limiting yourself to the most basic of tools, you become part of an ancient tradition that traces all the way back to the Egyptian arpedonapti, and corresponds intimately with Euclid's own methods. [My reference, a great read too!] That doesn't count for much in the actual math contest, but admit it, it's kind of cool.

Why you should freehand

• You're forced to focus only on what's given and where you want to go. No distractions, but you have to learn to ignore shaky lines and bad angles.
• You won't be misled by lines that are apparently concurrent, or triangles that are apparently similar in a constructed figure, but are not in general.
• Suppose you are a pure constructor, and the diagram is ugly: for example, you want the intersection of two lines, but in your figure, they're almost parallel and if you extend them the actual intersection is somewhere in outer space. (This happens awfully often in projective geometry problems.)
• Suppose you are a pure constructor, and for some reason you use a creaky compass that falls apart before you can even draw the first figure for what's probably the only geometry problem that's not monstrously difficult. (Of course, in some Olympiads I ended up bringing no less than three compasses, but you get the idea).
• There are some constructions that have been proven to be impossible using just a compass and a straightedge, squaring the circle, et cetera. Not that that's a relevant point for contests, though. It just rebuts the 'coolness factor' discussed above.

Of course I'm not saying each is a foolproof method in itself. Both freehanding and constructing are vulnerable to error that involve the act itself - the act of drawing a diagram in the first place. Configuration problems, for example. Now, there should be a way to deal with this. In theory, one should be able to make like a computer and abstractly view the problem as one regarding points as pure mathematical entities, but I guess to do Olympiad problems that way requires a mental capacity far greater than what most of us possess (but yes, I guess I have met people who have come close.).

So, should you construct, or should you freehand? You've probably guessed my verdict: the best method is the method you're comfortable with. I've met brilliant people from both of these "hallowed traditions", so whether you construct or freehand, you're in great company.

As for me? I don't do as much serious geometry now, and compasses aren't allowed in lots of places (plane cabins, for example). In short, I have been forced to put down the famed tool of the ancients. When I do geometry for Project Phi, I essentially freehand. So that ultimate question of my "allegiance" has been decided by simple convenience.

Before writing, however, I always make it a point to construct the figure in Geogebra (I still prohibit myself from using any measuring function), despite being fully aware that the beauty of my pen-on-paper figure couldn't possibly affect the rigor of the solution in any way. I don't know, I just need the comfort of a beautifully drawn diagram. Old habits die hard.