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In the list above, it is always taken to refer to non-intersecting lines. For example, if the word "parallel" in Playfair's axiom is taken to mean 'constant separation' or 'same angles where crossed by any third line', then it is no longer equivalent to Euclid's fifth postulate, and is provable from the first four (the axiom says 'There is at ...
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]
adventitious quadrangles problem. A quadrilateral such as BCEF is called an adventitious quadrangle when the angles between its diagonals and sides are all rational angles, angles that give rational numbers when measured in degrees or other units for which the whole circle is a rational number. Numerous adventitious quadrangles beyond the one ...
The easiest way to show this is using the Euclidean theorem (equivalent to the fifth postulate) that states that the angles of a triangle sum to two right angles. Given a line and a point P not on that line, construct a line, t, perpendicular to the given one through the point P, and then a perpendicular to this perpendicular at the point P.
The example shows trisection of any angle θ > 3π / 4 by a ruler with length equal to the radius of the circle, giving trisected angle φ = θ / 3 . Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.