A Curious Property of Octagons

Rogério César dos Santos;  Ana Clara Oliveira Comby;  Ramires Vargas da Silva

doi:10.32350/sir.32.01

A Curious Property of Octagons

Scientific Inquiry and Review ◽

10.32350/sir.32.01 ◽

2019 ◽

Vol 3 (2) ◽

pp. 01-07

Author(s):

Rogério César dos Santos ◽

Ana Clara Oliveira Comby ◽

Ramires Vargas da Silva

Keyword(s):

Plane Geometry ◽

Complex Numbers ◽

Analytical Geometry ◽

Famous Theorem ◽

Curious Property

The famous theorem of Van Aubel for quadrilaterals postulates that if squares are built externally on the sides of any quadrilateral, then the two segments that join the opposing centers of these squares are congruent and orthogonal. Inspired by this result and also by the results of Krishna, in this article we will prove the following result of plane geometry: each octagon is associated with a parallelogram, in some cases the parallelogram in question can be degenerate at a point or a segment. This is possible because of complex numbers and basics of analytical geometry.

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COMPLEX NUMBERS AND PLANE GEOMETRY

Invitation to Algebra ◽

10.1142/9789811219986_0005 ◽

2020 ◽

pp. 179-237

Keyword(s):

Plane Geometry ◽

Complex Numbers

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On rearrangements of infinite series

Proceedings of the Glasgow Mathematical Association ◽

10.1017/s2040618500033694 ◽

1958 ◽

Vol 3 (4) ◽

pp. 182-193 ◽

Cited By ~ 1

Author(s):

A. P. Robertson

Keyword(s):

Infinite Series ◽

Convergent Series ◽

Complex Numbers ◽

Famous Theorem

If a convergent series of real or complex numbers is rearranged, the resulting series may or may not converge. There are therefore two problems which naturally arise.(i) What is the condition on a given series for every rearrangement to converge?(ii) What is the condition on a given method of rearrangement for it to leave unaffected the convergence of every convergent series?The answer to (i) is well known; by a famous theorem of Riemann, the series must be absolutely convergent. The solution of (ii) is perhaps not so familiar, although it has been given by various authors, including R. Rado [7], F. W. Levi [6] and R. P. Agnew [2]. It is also given as an exercise by N. Bourbaki ([4], Chap. III, § 4, exs. 7 and 8).

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III. On scalar and clinant algebraical coordinate geometry, introducing a new and more general theory of analytical geometry, including the received as a particular case, and explaining ‘imaginary points,’ ‘intersections,’ and ‘lines.’

Proceedings of the Royal Society of London ◽

10.1098/rspl.1859.0076 ◽

1860 ◽

Vol 10 ◽

pp. 415-426 ◽

Cited By ~ 1

Keyword(s):

General Theory ◽

Present Theory ◽

Plane Geometry ◽

Analytical Geometry ◽

Abstract Equation ◽

Unit Of Length ◽

The Right ◽

Coordinate Geometry

Scalar Plane Geometry .— With O as a centre describe a circle with a radius equal to the unit of length. Let OA, OB be any two of its unit radii, termed ‘coordinate axes.’ From any point P in the plane AOB draw PM parallel to BO, so as to cut OA, produced either way if necessary, in M. Then there will exist some ‘scalars’ (‘real’ or ‘possible quantities’) u, v such that OM = u . OA, and Mp = v . OB, all lines being considered in respect both to magnitude and direction. Hence OP, which is the ‘appense’ or ‘geometrical sum’ of OM and MP, or = OM + MP, will = u . OA + v . OB. By varying the values of the 'coordinate scalars’ u, v P may be made to assume any position whatever on the plane of AOB. The angle AOB may be taken at pleasure, but greater symmetry is secured by choosing OI and OJ as coordinate axes, where IOJ is a right angle described in the right-handed direction. If any number of lines OP, OQ, OR, &c., be thus represented, the lengths of the lines PQ, QR, &c., and the sines and cosines of the angles IOP, POQ, QOR, &c., can be immediately furnished in terms of the unit of length and the coordinate scalars. If OP = x . OI + y . OJ, and any relation be assigned between the values of x and y , such as y = fx or ϕ ( x, y ) = 0 , then the possible positions of P are limited to those in which for any scalar value of x there exists a corresponding scalar value of y . The ensemble of all such positions of P constitutes the ‘ locus ’ of the two equations, viz. the ‘concrete equation’ OP = x . Ol + y .OJ, and the ‘abstract equation’ y = f. x. The peculiarity of the present theory consists in the recognition of these two equations to a curve, of which the ordinary theory only furnishes the latter, and inefficiently replaces the former by some convention respecting the use of the letters, whereby the coordinates themselves are not made a part of the calculation.

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