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Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2 π /5 radians (72° = 360°/5) can be trisected, but the angle of π /3 radians (60°) cannot be trisected. [8]
The angle π / 3 radians (60 degrees, written 60°) is constructible. The argument below shows that it is impossible to construct a 20° angle. This implies that a 60° angle cannot be trisected, and thus that an arbitrary angle cannot be trisected. Denote the set of rational numbers by Q.
Thus, an angle is constructible when = is a constructible number, and the problem of trisecting the angle can be formulated as one of constructing (). For example, the angle θ = π / 3 = 60 ∘ {\displaystyle \theta =\pi /3=60^{\circ }} of an equilateral triangle can be constructed by compass and straightedge, with x = cos θ ...
32-point compass rose. The points of the compass are a set of horizontal, radially arrayed compass directions (or azimuths) used in navigation and cartography.A compass rose is primarily composed of four cardinal directions—north, east, south, and west—each separated by 90 degrees, and secondarily divided by four ordinal (intercardinal) directions—northeast, southeast, southwest, and ...
The following construction is a variation of H. W. Richmond's construction. The differences to the original: The circle k 2 determines the point H instead of the bisector w 3. The circle k 4 around the point G' (reflection of the point G at m) yields the point N, which is no longer so close to M, for the construction of the tangent.
Construct a vertical line through the center. Mark one intersection with the circle as point A. Construct the point M as the midpoint of O and B. Draw a circle centered at M through the point A. Mark its intersection with the horizontal line (inside the original circle) as the point W and its intersection outside the circle as the point V.
A circle marked off in 15° angles is needed (circular protractor). An arbitrary point on the sub-style line is chosen. From here, a long-line, at right-angles to it, is drawn. A line is drawn at right angles from the substyle height, so that it passes through that point. Its length is noted. The length is copied from the point to O'.
It is possible to prove compass equivalence without the use of the straightedge. This justifies the use of "fixed compass" moves (constructing a circle of a given radius at a different location) in proofs of the Mohr–Mascheroni theorem, which states that any construction possible with straightedge and compass can be accomplished with compass alone.