Trisecting the angle
Here is a way to divide an angle into two equal smaller angles or, in other
words, bisect it. First place the compass point at O and, with any radius mark off
OA and OB. Moving the compass point to A, draw a portion of a circle. Do the
same at B. Label the point of intersection of these circles P, and with the straight
edge join O to P. The triangles AOP and BOP are identical in shape and therefore
the angles AÔP and BÔP will be equal. The line OP is the required bisector,
splitting the angle into two equal angles.
Can we use a sequence of actions like this to split an arbitrary angle into three
equal angles? This is the angle trisection problem.
If the angle is 90 degrees, a right angle, there is no problem, because the
angle of 30 degrees can be constructed. But, if we take the angle of 60 degrees,
for instance this angle cannot be trisected. We know the answer is 20 degrees but
there is no way of constructing this angle using only a straight edge and
compasses. So summarizing:
- you can bisect all angles all the time,
- you can trisect some angles all the time, but
- you cannot trisect some angles at any time.