Download 250 problems in elementary number theory by Waclaw Sierpinski PDF

By Waclaw Sierpinski

ISBN-10: 0444000712

ISBN-13: 9780444000712

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J 2 13. If a, b are the affixes of the vertices A, B of equilateral triangles ABC, ABC', w/3. find the affixes of C, C', knowing that angle (CA,CB) is Show that c + c' = a + b, cc' = a2 + b2 - ab, and also obtain these results from a relation of exercise 7. 14. The product of two inversions with the same center is a homothety. Show that the inversions are permutable only if their powers are opposite and that the homothety is then a symmetry. 15. Two inversions having distinct centers are permutable only if the square of the distance between the centers is the sum of the powers of inversion, or, in other words, only if the director circles or circles of double points are orthogonal.

If a, b are the affixes of the vertices A, B of equilateral triangles ABC, ABC', w/3. find the affixes of C, C', knowing that angle (CA,CB) is Show that c + c' = a + b, cc' = a2 + b2 - ab, and also obtain these results from a relation of exercise 7. 14. The product of two inversions with the same center is a homothety. Show that the inversions are permutable only if their powers are opposite and that the homothety is then a symmetry. 15. Two inversions having distinct centers are permutable only if the square of the distance between the centers is the sum of the powers of inversion, or, in other words, only if the director circles or circles of double points are orthogonal.

The equation of the product WlW2 of the rotations considered in the second example of article 21 being z' = zei(at,+at,) +a 2 (1 - ei at,) ei at, +a 1 (1- ei at,), proves, if we compare with equation (17), that the points Z', Z2 coincide only if we have a 1 (l - ei at,) ei at, + a2 (1 - ei at,) = a 2 (1 - ei at,) ei at, + a 1 (1 - ei at,) 24. ) o. ) = Since

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