Angle trisection is one of the classic problems in mathematics. It is an extremely essential factor concerning the precision of calculations. Human being has not achieved finding a general solution for that by this day. Since ancient times, it has been the aspiration of every mathematician to be able to trisect an arbitrary angle. Nevertheless high profile contemporaries have closed the door on this crucial problem by assuming it impossible based on weak inductions, boosted by inability in finding a solution; bare theories without a slight clue regarding an authority over straightedge and compass constructions.(1)
In the circle with the center S, we draw the two perpendicular diameters and the bisector(2) of the ninety degree angle GSF and extend it so that it crosses the circumference of the circle at the other end.
Using the solid compass, we divide the ninety degree arc GF into three equal thirty degree arcs by sectioning the sixty degree arcs GK and FP. Therefore the three central adjacent angles are equal to one another.
Now we connect the points G and F to the points Z and W oppositely. An internal sixty degree angle is formed; the angle known to be impossible to trisect. However, using the trisection points P and K, the sixty degree angle will be trisected flawlessly.
Supposing a pair of arcs with the ratio of x between them, there exist countless pairs of arcs with the same ratio on the circumference of the circle accordingly. This evident fact is due to the constancy of the curvature and the fact that circle is a locus (central symmetry) in combination with the quality of the diameter (axial symmetry).The three central adjacent angles GSP, PSK and KSF are equal to one another. We connect the point M to the points G, P, K and F. The inscribed adjacent angles GMP, PMK and KMF are formed which are also equal to one another by the measure of one half of the intercepted arcs. The three adjacent angles are equal at the center and on the circumference. Therefore the countless internal angles enclosed in between the two trisected central and inscribed angles with the common intercepted arcs, have to be trisected as long as the vertex stays on the axis of symmetry. This is also the case for the external angles.
Central and inscribed angles are special kinds of internal angles with two intercepted arcs. We simply avoid redundant addition and division in those cases by taking one equal to the intercepted arc and the other one a half of the intercepted arc. The measure of the external angle is also a half of the variation of the two arcs.
It can not possibly be imagined that someone acquainted with the basic principles in geometry, declines the foundation of mathematics by rejecting these sorts of reasoning. The underlying principles of geometry are more reliable than any other deficient derivative method. Any theory conflicting with a principal theorem is invalid.(3)
The proportions of the adjacent angles are constantly retained through shifting the vertex along the axis of symmetry.
(1) For further clarification, you may refer to the article “A Shot In The Dark”.
(2) the locus on which all points have the same quality
(3) geometrical reasoning based on numeric methods