******** cubic_parabola_tri_******** You can see the process in animation. Tomasso Ceva(1648 - 1737) ,brother of Giovanni Ceva(1647 - 1734) ,who is known for Ceva's theorem, applied "Insertion method" by Archimedes to Angle trisection using a special curve called "Cycloidum anomalarun", or "The Cycloid of Ceva".
So once the curve is drawn,the trisection process is straight forward. For detail, go to the section Trisectrix - Mac Laurin.
For detail, go to the section Quadratrix - Hippias. Pick a point "A" on the outer circle to define angle AOB for trisection.
Trisection using Conchoid is shown in the figure shown below. Get intersecting point "P", and drop the perpendicular to OB.
One cannot square any circle, nor can one double any cube. For example there is a fairly straightforward method to trisect a right angle.
For given the right angle CAB draw a circle to cut AB at E.
It is an easy task to tell that a 'proof' one has been sent 'showing' that the trisector of an arbitrary angle can be constructed using ruler and compasses must be incorrect since no such construction is possible.
Firstly it has no real history relating to the way that the problem first came to be studied.
Secondly it is a problem of a rather different type.
For detail, go to the section Conchoid - Nicomedes. ******** limason_tri_******** You can see the process in animation. Rene Descartes(1596 - 1650) ,who is called the founder of the "Analytic Geometry",published the famous treatise "La Geometrie" in 1637. Draw a circle with radius=2,then define a point A .
Archimedes(287 BC - 212 BC) used Spiral (called Archimedes' Sprial) for Trisection. ************* spiral_tri_************* You can see the process in animation. Pappus(280 - 350) showed that "Huyperbola" can be used for Angle Trisection. The trisection of angle F1-O-F2 is equivalent to dividing arc F1-F2 into 3 equal arc length. The type of Limaçon used for Trisection is b = 1 case. Select a point "A" on the Limaçon to define an angle AOB to be trisected. In this book he showed that Angle Trisection can be done by using Parabola.
This curve was first called "Cochloid", then later named "Conchoid". This will cut the circle drawn at step 3 at point "T" 5. ******** parabola_tri_******** You can see the process in animation.