I have investigated a series of geometric phenomena that may allow the seven and multi-division of the circle with straight edge and compass as part of my integral arts practice. I have tried to do the necessary calculations and I have come up with accuracy up to two decimals. (I started with three decimals and I imagine that the slight variance on the third decimal is due to the rounding off effect). Below are the constructions and proofs I have been able to produce.
Figure 1 and Figure 2 on page 1 show the two well-known phenomena of Incentre of a triangle and the triangle Centroid. Figure 3 and Figure 4 show two closely related phenomena, which have hitherto gone noticed, or if noticed, unutilized
In Figure 3 the three angles AOC, COB, BOA of the rectangular triangle ABC have been divided into three.
G and F are the trisection points between points A and C.
Point S is the midpoint between G and F.
E and D are the trisection points between points B and C.
Point R is the midpoint between E and D.
H and I are the trisection points between points A and B.
Point S is the midpoint between H and I.
If lines are drawn from point S to point B, from point R to point A and from point C to point T all three lines seem to intersect in one point P.
The same applies to Figure 4, where the same process is applied to the division of each of the three angles of a scalene triangle into seven parts.
PAGE 2 & PAGE 3 are the sketch and the mathematical proof for Figure 3.
PAGE 4 & PAGE 5 are the sketch and the mathematical proof for Figure 4.
If the above-mentioned lines really meet in one point P, then the seven-, nine- and indeed multi-division of the circle with straight edge and compass is possible. I have demonstrated the process of this division on PAGE 6.
If the above assumptions are true, the seven-division (as well as all other divisions) of the circle can be done in the following way:
Chose a random length AD (approximating one side of a scalene triangle) and draw it seven times on a circle. The result is the angle AOC divided in seven parts. The points that mark the middle section are E and F.
Half the angle AOC to construct line l. Line l intersects line EF in the midpoint R.
Now take a different random length and repeat the same process to obtain the angle COB divided into seven. The points that mark the middle section are H and I.
Half the angle COB to construct line m. Line m intersects line HI in the midpoint S.
Half the angle BOA to construct line n.
Connect point R with point B and point S with point A. The two lines will intersect in point P.
If a line is drawn from point C through point P it will intersect with line n in point T.
Then a line is drawn through point T that is at a right angle to line n. This line will intersect the circle in the two points W and V. These two points are 1/7 of the angle AOB.
Figure 6 & Figure 7
We have obtained three times seven divisions of the circle. If we add 1/7 of each of the three angles the result must be 1/7 of the circle. If this is so the circle can be divided into seven; the same method can be used to divide the circle into any number of divisions with straight edge and compass.