ANGLE TRISECTION

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. At this moment, those theories are encountered with a critical challenge in confrontation with the foundational geometrical theorems and principles.(1)

We draw the circle with the center S and a desired radius. We draw the two perpendicular diameters FR and GT and the bisector of the ninety degree arc GSF and extend it so that it crosses the circumference of the circle at the points R and M.

We draw the chord GF so that the bisector is crossed at the point H. The central adjacent angles GSR and FSR are equal to one another and the intercepted arcs.2We connect the point M to the points G and F. The inscribed adjacent angles GMR and FMR are formed which are also equal to one another with the measure of a half of the intercepted arcs.

We suppose the desired point D on the bisector (the axis of symmetry) and connect that to the points G and F. The internal adjacent angles GDR and FDR are also equal to each other since the two right triangles GDH and FDH are equal one another. This is also the case for the external adjacent angles GNR and FNR with the vertex outside of the circle.

It is a very important and noticeable point that not only the central and the inscribed adjacent angles with equal intercepted arcs are equal to each other, also they are equal to one another as long as the vertex is anywhere on the axis of symmetry (the bisector of the sum of the adjacent angles).

The following three significant principles are concluded at this point:

A – The equal adjacent angles intercept equal arcs as long as the vertex stands on the axis on symmetry.

B – The adjacent angles intercepting equal arcs are equal to one another as long as the vertex stands on the axis of symmetry.

C – The proportions of the adjacent angles are constantly retained through shifting the vertex along the axis of symmetry.

We draw the circle with the center S and a desired radius. We draw the two perpendicular diameters GT and FR and the bisector of the ninety degree arc GSF and extend it so that it crosses the circumference of the circle at the points R and M.

Using the solid compass, we divide the ninety degree chord GF into three equal thirty degree arcs GP, KP and KF. The three central adjacent angles GSP, PSK and KSP are equal to one another. Now we suppose the desired point D on the axis of symmetry(2) and connect that to the points G, P, K and F. The measure of the internal angle GDP is twice as that of the internal angle PDR since they are facing intercepted arcs with the ratio of two to one.3Therefore we can conclude that the two internal adjacent angles GDP and PDK are equal to one another. Thereby the three internal adjacent angles GDP, PDK and KDF are equal to one another. This is also the case for the external adjacent angles with their vertex at the point N.

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.

We can also look at it this way: The measure of the angle GDP constantly equals that of the arc GP plus that of the arc YZ divided by two, which is the same as the arc PK plus the arc XY divided by two, which is also the same as the arc KF plus the arc WX divided by two.511Centering the point D and rotating the two lines FW and GZ until they superpose on the lines KX and PY, both of the arcs GF and WZ loose two thirds of their measure. Considering a half of the sum of the two arcs, the vertically opposite sixty degree angles are divided proportionally.(3)

Supposing a pair of arcs with the ratio of x between them, there exist countless pairs of arc with the same ratio on the circumference of the circle. This is due to the constancy of the curve of the circle which is a locus (central symmetry) in combination with the quality of the diameter (axial symmetry).2526Ultimately we can reason this way as well: In the next figure, as we shift the vertex on the smaller concentric circle, the arc x is greater than the arc y on the right side and the arc z on the left side. It is also smaller than the arc z on the right side and the arc y on the left side.

As we work our way towards the middle, the variation rate decreases point by point until the proportions are inverted passed a certain point on which the measure of the arc x equals that of the arc y from one side and that of the arc z from another side. For the inversion to take place, there certainly must exist a turning point on which the three arcs equal which is exclusively on the axis of symmetry.(4)27Supposing the arc x greater or smaller than either of the arcs y and z while the vertex is on the axis of symmetry, it contradicts the mentioned proportions considering an epsilon to the right or an epsilon to the left. In other words, a specific measure can not be greater and smaller than another measure simultaneously.(5)28Discrete inconsistent contraction rate of the angles, would contradict the continuity of the arc of the circle and the fact that circle is a locus and the diameter is the axis of symmetry. Besides, the arc x itself, consists of countless points representing countless internal adjacent angles to those the same rule apply as well.(6)

It can not possibly be imagined that someone acquainted with the basic principles in mathematics, rejects these sorts of reasoning. The underlying principles of geometry are more reliable than any other deficient derivative method, now confronted with a serious challenge since the classic problem of angle trisection is generally solved, using straightedge and compass. Any theory conflicting with a principal theorem is invalid.(7)

The proportions of the adjacent angles are constantly retained through shifting the vertex along the axis of symmetry.

 Animated Demonstration


(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) 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.

(4) On the other hand, the three adjacent angles are inscribed in the smaller concentric circle. The measure of the intercepted arcs are equal only when the vertex is on the diameter so that the three inscribed angles would be equal to one another by one half of the intercepted arcs.

(5) proof by contradiction

(6) The theorem is not restricted to three arcs. We can suppose as many arcs as desired on the circumference of the circle in the same order. Accordingly division of an arbitrary angle into five or seven equal sections would be possible.

(7) geometrical reasoning based on numeric methods

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