Editing Angle trisection (section)

{{Short description|Construction of an angle equal to one third a given angle}}

[[Image:Neusis-trisection.svg|thumb|right|Angles may be trisected via a [[neusis construction]] using tools beyond an unmarked straightedge and a compass. The example shows trisection of any angle {{math|''θ'' > {{sfrac|3π|4}}}} by a ruler with length equal to the radius of the circle, giving trisected angle {{math|''φ'' {{=}} {{sfrac|''θ''|3}}}}.]]

'''Angle trisection''' is a classical problem of [[straightedge and compass construction]] of ancient [[Greek mathematics]]. It concerns construction of an [[angle]] equal to one third of a given arbitrary angle, using only two tools: an unmarked [[straightedge]] and a [[Compass (drawing tool)|compass]].

In 1837, [[Pierre Wantzel]] proved that the problem, as stated, is [[Proof of impossibility|impossible]] to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a [[right angle]].

It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, [[neusis construction]], also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries.

Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of [[pseudomathematics|pseudomathematical]] attempts at solution by naive enthusiasts. These "solutions" often involve mistaken interpretations of the rules, or are simply incorrect.<ref name="trisectors"/>