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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]
A diagram of an object in two plane mirrors that formed an angle bigger than 90 degrees, causing the object to have three reflections. A plane mirror is a mirror with a flat reflective surface. [1] [2] For light rays striking a plane mirror, the angle of reflection equals the angle of incidence. [3]
The next construction is originally due to Archimedes, called a Neusis construction, i.e., that uses tools other than an un-marked straightedge. The diagrams we use show this construction for an acute angle, but it indeed works for any angle up to 180 degrees. This requires three facts from geometry (at right):
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 ...
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.
Bisection of an angle using a compass and straightedge. An angle bisector divides the angle into two angles with equal measures. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle. The 'interior' or 'internal bisector' of an angle is the line, half-line, or line segment that
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Proof: Construct a sphere on top of each circle and then construct a plane tangent to these three spheres. The plane intersects the plane that the circles lies on at a straight line containing P 1, P 2, and P 3. These points are also the centers of homothety for the circles that they were derived from. [4]