By Sharipov R.A.
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Extra resources for Foundations of geometry for university students and high school students (Ufa 1998)
Example text
Prove that each tetrahedron ABCD is the union of its interior and the triangles ABC, ABD, ACD, and BCD. CHAPTER III AXIOMS OF CONGRUENCE. § 1. Binary relations of congruence. The axioms of congruence form the third group of Euclid’s axioms. In formulating these axioms it is assumed that in the set of all straight line segments a binary relation is defined which is called the congruence. A similar binary relation is assumed to be given in the set of all angles. It is also called the congruence, though the congruence of segments and the congruence of angles are certainly two different binary relations.
If X and Y are in different parts of this partitioning, then the interval (XY ) contains at least one of the points A or C. 4). 2 after that is proved by applying the first item from the axiom A15. Now let’s consider the case where the points X and Y lie on the segment [AC]. Then X ≺ Y implies A ≺ X ≺ Y ≺ C. Applying the axiom A13, we choose a point Y˜ on the ray [f (X)M such that [f (X)Y˜ ] ∼ = [XY ]. Then, using the same axiom A13, we draw the ˜ ∼ segment [Y˜ C] = [Y C] on the ray coming out from the point Y˜ in the direction opposite to the ray [Y˜ K .
Assume that the point B lies between A and C, while the point C lies between B and D. Then both points B and C lie between the points A and D. Proof. From (A ◮ B ◭ C) it follows that the point A lies on the line BC, while from (B ◮ C ◭ D) it follows that D also lies on the line BC. 1 all of the four points A, B, C, and D lie on one straight line. 2 we find a point E not lying on the line AD (see Fig. 1). Then we apply the axiom A10 to the points C and E. As a result on the line CE we find a point F such that the point E lies in the interior of the segment [CF ].