Abstract
The constructions of three classical Greek problems (squaring the circle, doubling the cube and angle trisection) using only a ruler and a compass are considered unsolvable. The aim of this article is to explain the original methods of construction of the above-mentioned problems, which is something new in geometry. For the construction of squaring the circle and doubling the cube the Thales' theorem of proportional lengths has been used, whereas the angle trisection relies on a rotation of the unit circle in the Cartesian coordinate system and the axioms of angle measurement. The constructions are not related to the precise drawing figures in practice, but the intention is to find a theoretical solution, by using a ruler and a compass, under the assumption that the above-mentioned instruments are perfectly precise.
Highlights
C and doubling the cube the Thales' theorem of proportional lengths has been used, whereas the angle trisection relies on a rotation of the unit circle in the Cartesian coordinate system and the axioms of angle
We shall prove that the area of the given circle k(O, r) equals the area of the square ADGH (Fig. 3)
We shall calculate the area of the square ADGH
Summary
In everyday speech we may hear the expression “squaring the circle” used as a metaphor for trying to solve something impossible. With a compass and a straightedge, we construct the line n perpendicular to the length AB through the point C and denote its intersection with the semicircle by the point D. As shown by the previous method, when constructing the length X= 2 , we divide the diameter AB by the point C in the ratio of integers 11000000 and 3005681, i.e. AC : CB = 11000000 : 3005681, in the following way: On the arbitrary ray Aq, we determine the point M by “transferring” 11000000 arbitrary unit lengths. Through the point C we construct the line l so that it is parallel to the ray Aq and its intersection with the length NB we denote by the point L (Fig. 3).
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