trisection of angle
Given an angle of measure (http://planetmath.org/AngleMeasure) α such that 0<α≤π2, one can construct an angle of measure α3 using a compass and a ruler (http://planetmath.org/MarkedRuler) with one mark on it as follows:
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1.
Construct a circle c with the vertex (http://planetmath.org/Vertex5) O of the angle as its center. Label the intersections
of this circle with the rays of the angle as A and B. Mark the length OB on the ruler.
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2.
Draw the ray →AO.
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3.
Use the marked ruler to determine C∈c and D∈→AO such that CD=OB and B, C, and D are collinear
. Draw the line segment
¯BD. Then the angle measure of ∠CDO is α3. (The line segment ¯OC is drawn in red. Having this line segment drawn is useful for reference purposes for the justification of the construction.)
Let m denote the measure of an angle. Then this construction is justified by the following:
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•
Since ∠AOB is an exterior angle
of △BOD, we have that m(∠AOB)=m(∠OBD)+m(∠ODB);
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•
Since OC=OB=CD, we have that △BOC and △OCD are isosceles triangles
;
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•
Since the angles of an isosceles triangle are congruent
, m(∠OBC)=m(∠OCB) and m(∠COD)=m(∠CDO);
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•
Since ∠OCB is an exterior angle of △OCD, we have that m(∠OCB)=m(∠COD)+m(∠CDO);
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•
Note that ∠OBC=∠OBD and ∠ODB=∠CDO;
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•
Thus,
α=m(∠AOB)=m(∠OBD)+m(∠ODB)=m(∠OBC)+m(∠CDO)=m(∠OCB)+m(∠CDO)=m(∠COD)+m(∠CDO)+m(∠CDO)=3m(∠CDO).
Note that, since angles of measure π6, π3, and π2 are constructible using compass and straightedge, this procedure can be extended to trisect any angle of measure β such that 0<β≤2π:
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•
If 0<β≤π2, then use the construction given above.
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•
If π2<β≤π, then trisect an angle of measure β-π2 and add on an angle of measure π6 to the result.
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•
If π<β≤3π2, then trisect an angle of measure β-π and add on an angle of measure π3 to the result.
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•
If 3π2<β≤2π, then trisect an angle of measure β-3π2 and add on an angle of measure π2 to the result.
This construction is attributed to Archimedes.
References
- 1 Rotman, Joseph J. A First Course in Abstract Algebra. Upper Saddle River, NJ: Prentice-Hall, 1996.
Title | trisection of angle |
---|---|
Canonical name | TrisectionOfAngle |
Date of creation | 2013-03-22 17:16:35 |
Last modified on | 2013-03-22 17:16:35 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 11 |
Author | Wkbj79 (1863) |
Entry type | Algorithm |
Classification | msc 01A20 |
Classification | msc 51M15 |
Related topic | VariantsOnCompassAndStraightedgeConstructions |