trisection of angle
Given an angle of measure (http://planetmath.org/AngleMeasure) such that , one can construct an angle of measure 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 with the vertex (http://planetmath.org/Vertex5) of the angle as its center. Label the intersections of this circle with the rays of the angle as and . Mark the length on the ruler.
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2.
Draw the ray .
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3.
Use the marked ruler to determine and such that and , , and are collinear. Draw the line segment . Then the angle measure of is . (The line segment is drawn in red. Having this line segment drawn is useful for reference purposes for the justification of the construction.)
Let denote the measure of an angle. Then this construction is justified by the following:
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Since is an exterior angle of , we have that ;
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Since , we have that and are isosceles triangles;
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Since the angles of an isosceles triangle are congruent, and ;
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Since is an exterior angle of , we have that ;
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Note that and ;
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Thus,
Note that, since angles of measure , , and are constructible using compass and straightedge, this procedure can be extended to trisect any angle of measure such that :
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If , then use the construction given above.
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If , then trisect an angle of measure and add on an angle of measure to the result.
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If , then trisect an angle of measure and add on an angle of measure to the result.
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If , then trisect an angle of measure and add on an angle of measure 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 |