Deductive Geometry

E.A. Maxwell

• Editore: Dover Publications
• Anno: 2016
• ISBN:
  9780486809250
• Argomento: Mathematik
• Categoria: Geometrie

This concise review examines the geometry of the straight line, circle, plane, and sphere as well as their associated configurations, including the triangle and the cylinder. Aimed at university undergraduates, the treatment is also useful for advanced students at the secondary level.
The straightforward approach begins with a recapitulation of previous work on the subject, proceeding to explorations of advanced plane geometry, solid geometry with some reference to the geometry of the sphere, and a chapter on the nature of space, including considerations of such properties as congruence, similarity, and symmetry. The text concludes with a brief account of the elementary transformations of projection and inversion. Numerous examples appear throughout the book.

THREE

Some Circle Theorems

1. The Theorem of Ptolemy

Fig. 23

(i) A quadrilateral inequality

THE PROBLEM. ABC is a given triangle and D an arbitrary point. (The diagram shows a typical position of D; other cases are possible, but the differences are slight.) A triangle ABU is constructed as in the diagram so that

We are to prove that

and hence that

THE DISCUSSION.

Also

Hence, in ΔACU, ΔADB,

and

so that

Moreover, from the similar triangles,

and

Hence

(ii) The cyclic case; Ptolemy’s theorem.

The final step of the preceding work,

presupposed that U was not on BC, a condition that would certainly hold in general. The case of exception must now receive attention.

THE PROBLEM. Suppose that, in the preceding work, U ∈ BC. Then

so that

It is required to prove that

[Note the double arrow ⇔.]

Suppose, first, that equality holds, so that U ∈ BC. Then

Hence A, B, C, D are concyclic.

Suppose, next, that A, B, C, D are concyclic. Then

so that BU, BC are the same lines.

Hence U ∈ BC, so that the equality holds.

Note. The four points A, B, C, D can be split in two pairs in three ways,

Ptolemy’s theorem asserts that, when the four points are con-cyclic, the sum of products from two of these pairs is equal to the third. The pair which comes “third” is that defined by the diagonals of the cyclic quadrilateral.

[For a more detailed discussion of implications, see E. A. Maxwell, Fallacies in Mathematics, Cambridge University Press (1959) p. 28.]

Problems

1. If D is a point on the arc opposite A of the circumcircle of an equilateral triangle ABC, then AD = BD + CD.

2. AB, PQ are parallel chords of a circle. Prove that

3. Identify the well-known result which is a special case of the theorem of Ptolemy when ABCD is a rectangle.

4. In ΔABC, draw BE ⊥ AC, CF ⊥ AB. Prove that BF = BC cos B, CF = BC sin B, and write down similar expressions for CE, BE.

By applying the theorem of Ptolemy ...