Trisecting the Angle

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Classic Problem

Trisecting the angle is dividing a given angle into three equal parts..


Solution

Trisecting the Angle by Compass and Straightedge Construction is Impossible.


Various techniques were devised which used more elaborate apparatus than just the straightedge and compass:


Archimedean Spiral

Let $\alpha$ be an angle which is to be trisected.

This can be achieved by means of an Archimedean spiral.


Neusis Construction

Let $\alpha$ be an angle which is to be trisected.

This can be achieved by means of a neusis construction.


By Parabola

Let $\alpha$ be an angle which is to be trisected.

This can be achieved by means of a parabola.


By Hyperbola

Let $\alpha$ be an angle which is to be trisected.

This can be achieved by means of a hyperbola.


Quadratrix of Hippias

Let $\alpha$ be an angle which is to be trisected.

This can be achieved by means of a quadratrix of Hippias.


Cissoid of Diocles

Let $\alpha$ be an angle which is to be trisected.

This can be achieved by means of a cissoid of Diocles.


Conchoid of Nicomedes

Let $\alpha$ be an angle which is to be trisected.

This can be achieved by means of a conchoid of Nicomedes.


Fallacious Proofs

Edw. J. Goodwin

The trisection of a right line taken as the chord of any arc of a circle trisects the angle of the arc.
-- Edw. J. Goodwin


Also see


Historical Note

The exercise to trisect of the general angle using a compass and straightedge construction, was an exercise that the ancient Greeks failed to succeed in.

This was one of three such problems: the other two being Squaring the Circle and Doubling the Cube.

There are several techniques available that use other tools, but these were considered unacceptably vulgar to the followers of Plato.


Sources