Trisecting the Angle

The ancient Greeks conjectured about it. Modern mathematicians proved that it can't be done. Here's an interesting approximation.

Background: In high-school, I conjectured that if you could trisect a line, you could use that trisection to trisect an angle. Where did I make my error?

Written 2000

Formatted 2010

  The Method:

Step 1: Start with an angle and draw a line across that angle.

trisect step 2

Step 2: Triple that line by replicating it twice. (Replication of a line is a well known construction for which all the steps will not be shown.)

trisect step 3

Step 3: Translate this new line, using perpendiculars, to where its ends meet the angle.

trisect step 4

Step 4: You now have a trisected line crossing an angle. Use the line segments to trisect the angle.

trisect step 5

Since we know it can't be done, what is the error in our method? It is our original assumption that by trisecting a line we can trisect an angle. If we show the lines that really must be equal to trisect an angle we can show that our method is only an approximation. Challenge for students: How good of an approximation is this?


trisect step 1


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  Method 2:

Why can't we use the tripling idea with angles instead of lines?

That is, make an arbitrary angle and triple it by copying it twice using the well known construction for copying angles.

error fix

Our trisected angle must be placed over the angle we want to trisect.

error fix

Can we now find a way to squeeze our arbitrary angle until it matches our original angle?

Mathematicians say, "No." But, why?

Addendum: I have not had a chance to review the proof that an angle can not be trisected (Ref 1). I hear that the proof involves using algebra to describe all the possible construction processes with equations, and then determining what changes in slope can be produced with those equations.

If you think you have developed a method to trisect an angle try using algebra to describe your method. Odds are you will find that your method requires you to trisect one angle in order to trisect the second. That is you can't trisect until you have already trisected - a contradiction.


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