Angle trisection

Angle trisection is a classical problem of <a href="/facts/Straightedge_and_compass_construction/lbaHf2m7">straightedge and compass construction</a> of ancient <a href="/facts/Greek_mathematics/ealke3z8">Greek mathematics</a>. It concerns construction of an <a href="/facts/Angle/gFUTEQYq">angle</a> equal to one third of a given arbitrary angle, using only two tools: an unmarked <a href="/facts/Straightedge/6o9Xp9Eb">straightedge</a> and a <a href="/facts/Compass_(drawing_tool)/0uNSeGy9">compass</a>.
In 1837, <a href="/facts/Pierre_Wantzel/gKVItq1G">Pierre Wantzel</a> proved that the problem, as stated, is <a href="/facts/Proof_of_impossibility/rAQEghH8">impossible</a> to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a <a href="/facts/Right_angle/psaslRq7">right angle</a>.
It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, <a href="/facts/Neusis_construction/teuATjxg">neusis construction</a>, 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 <a href="/facts/Pseudomathematics/emTuA5AE">pseudomathematical</a> attempts at solution by naive enthusiasts. These "solutions" often involve mistaken interpretations of the rules, or are simply incorrect.

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Angle trisection