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A new point is determined in Euclidean Geometry exclusively in one of the three following ways:
Having given four points A, B, C, D, not all incident on the same straight line, then
Whenever a point P exists which is incident both on and on , that point is regarded as determinate.
Whenever a point P exists which is incident both on the straight line and on the circle C, that point is regarded as determinate.
Whenever a point P exists which is incident on both the circles A, C, that point is regarded as determinate.
The cardinal points of any figure determined by a Euclidean construction are always found by means of a finite number of successive applications of some or all of these rules , and . Whenever one of these rules is applied it must be shown that it does not fail to determine the point. Euclid's own treatment is sometimes defective as regards this requisite.
In order to make the practical constructions which correspond to these three Euclidean modes of determination, correponding to the ruler is required, corresponding to both ruler and compass, and corresponding to the compass only.
...it is possible to develop Euclidean Geometry with a more restricted set of postulations. For example it can be shewn that all Euclidean constructions can be carried out by means of alone... (en) |
