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A treatise on the analytical geometry of the point, line, by Casey J. PDF

By Casey J.

ISBN-10: 1418182842

ISBN-13: 9781418182847

This quantity is made from electronic photographs created in the course of the college of Michigan collage Library's upkeep reformatting software.

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Extra info for A treatise on the analytical geometry of the point, line, circle, and conical sections (1885)

Example text

Lemma is proved. 1 proved just above the relation of congruence is an equivalence relation in the set of all straight line segments. Axiom A15. Let B be a point lying between A and C on a straight line AC, while L be a point lying between K and M on a straight line KM . Then the following propositions are valid: (1) [AB] ∼ = [KL] and [BC] ∼ = [LM ] imply [AC] ∼ = [KM ]; ∼ ∼ (2) [AB] = [KL] and [AC] = [KM ] imply [BC] ∼ = [LM ]. Note that the propositions (1) and (2) under the assumptions of the axiom A15 can be complemented with one more proposition of the same sort: (3) [AC] ∼ = [KM ] and [BC] ∼ = [LM ] imply [AB] ∼ = [KL].

Now we apply the axiom A13 to the segment [AB] and to the ray [AB . It says that the point E on the ray [AB such that [AB] ∼ = [AE] is unique. But [AB] ∼ = [AB]. Therefore, the point E coincides with B. Hence, [CD] ∼ = [AB]. Lemma is proved. 1 proved just above the relation of congruence is an equivalence relation in the set of all straight line segments. Axiom A15. Let B be a point lying between A and C on a straight line AC, while L be a point lying between K and M on a straight line KM . Then the following propositions are valid: (1) [AB] ∼ = [KL] and [BC] ∼ = [LM ] imply [AC] ∼ = [KM ]; ∼ ∼ (2) [AB] = [KL] and [AC] = [KM ] imply [BC] ∼ = [LM ].

The angle produced as the intersection of the closed half-planes a+ and b+ is usually denoted as follows: ∠AOB = a+ ∩ b+ . 3) Applying the axiom A10, now we choose a point C on the line a such that the point O lies between A and C. A point D on the line b is chosen in a similar way. The lines a and b define four angles at a time on the plane α: ∠AOB = a+ ∩ b+ , ∠BOC = a+ ∩ b− , ∠COD = a− ∩ b− , ∠DOA = a− ∩ b+ . The points A and B marking the half-planes a+ and b+ play equal roles in defining the angle ∠AOB.

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A treatise on the analytical geometry of the point, line, circle, and conical sections (1885) by Casey J.


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