
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 highschool, 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.


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.)


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


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


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?



Related pages at this site


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.


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


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.





