theorem on constructible angles
Let . Then the following are equivalent:
An angle of measure (http://planetmath.org/AngleMeasure) is constructible (http://planetmath.org/Constructible2);
is a constructible number;
is a constructible number.
First of all, due to periodicity, we can restrict our attention to the interval . Even better, we can further restrict our attention to the interval for the following reasons:
If an angle whose measure is is constructible, then so are angles whose measures are , , and ;
If is a constructible number, then so is .
If , then clearly an angle of measure is constructible, and . Thus, equivalence (http://planetmath.org/Equivalent3) has been established in the case that . Therefore, we can restrict our attention even further to the interval .
Assume that an angle of measure is constructible. Construct such an angle and mark off a line segment of length from the vertex (http://planetmath.org/Vertex5) of the angle. Label the endpoint that is not the vertex of the angle as .
Now assume that is a constructible number. At one endpoint of a line segment of length , erect the perpendicular to the line segment.
From the other endpoint of the given line segment, draw an arc of a circle with radius so that it intersects the erected perpendicular. Label this point of intersection as . Connect to the endpoint of the line segment which was used to draw the arc. Then an angle of measure and a line segment of length have been constructed.
A similar procedure can be used given that is a constructible number to prove the other two statements. ∎
Note that, if , then any of the three statements thus implies that is a constructible number. Moreover, if is constructible, then a right triangle having a leg of length and another leg of length is constructible, which implies that the three listed conditions are true.
|Title||theorem on constructible angles|
|Date of creation||2013-03-22 17:15:59|
|Last modified on||2013-03-22 17:15:59|
|Last modified by||Wkbj79 (1863)|