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Let f = X 6 − 3 X 2 − 1 ∈ Q [ X ] and let L f be the splitting field of f over Q . We show by hand that the Galois group Gal (L f / Q) of the Galois extension L f / Q is isomorphic to the alternating group A4. Moreover, we show that the six roots of f correspond to the six edges of a tetrahedron and that the four roots of the polynomial X 4 + 18 X 2 − 72 X + 81 correspond to the four faces of a tetrahedron, which allows us to determine all eight proper intermediate fields of the extension L f / Q . [ABSTRACT FROM AUTHOR]

We give an algebraic proof of a slightly extended version of Morley's Five Circles Theorem. The theorem holds in all Miquelian Möbius planes obtained from a separable quadratic field extension, in particular in the classical real Möbius plane. Moreover, the calculations bring to light a hidden twin of the Five Circles Theorem that seems to have been overlooked until now. [ABSTRACT FROM AUTHOR]

Recent progress in the theory of Heron triangles and their elliptic curves led to new families of congruent number elliptic curves with rank at least two. Based on these results, we derive a parametric family of congruent number elliptic curves with rank at least three. It turns out that this family is isomorphic to a family which was recently discovered by the third-named author, however the new approach is simpler, more flexible and gives new insight. In particular, it provides in addition three formulae for congruent numbers. [ABSTRACT FROM AUTHOR]

Let F be a set of functions with a common domain X and a common range Y. A set S ⊆ X is called a set of range uniqueness (SRU) for F , if for all f , g ∈ F , f [ S ] = g [ S ] ⇒ f = g. Let P n , k be the set of all real polynomials in n variables of degree at most k and let L k (R n , R n) be the set of all linear functions f : R n → R n with rank k. We show that there are SRU's for P n , k of cardinality 2 n + k k − 1 , but there are no such SRU's of size 2 n + k k − 2 or less. Moreover, we show that there are SRU's for L k (R n , R n) of size 2 n − 1 i f k = 1 , 2 n − k + 1 i f k > 1 , but there are no such SRU's of smaller size. [ABSTRACT FROM AUTHOR]

More than a century ago, Norman Anning conjectured that it is possible to arrange 48 points on a circle, such that all distances between the points are integer numbers and are all among the solutions of the diophantine equation x2 + xy + y2 = 72 · 132 · 192 · 312. We shall obtain Anning's conjecture as a consequence of a far more general geometrical result. [ABSTRACT FROM AUTHOR]

More than a century ago, Norman Anning conjectured that it is possible to arrange 48 points on a circle, such that all distances between the points are integer numbers and are all among the solutions of the diophantine equation x2 + xy + y2 = 72 · 132 · 192 · 312. We shall obtain Anning's conjecture as a consequence of a far more general geometrical result. [ABSTRACT FROM AUTHOR]

The article discusses proof of Poncelet's theorem using real projective plane based on Pascal's theorem. Topics discussed include Poncelet's theorem related to n-sided polygon in nondegenerate conics and vertices, intersections of circles, Abelian integrals theory, interpretation of elliptic billiards, Poncelet's theorem being consequence of Pascal's theorem and a theorem proving sides of triangles being tangent if triangles are inscribed in a conic.

Focuses on the generalization of the arithmetic progressions in a smooth Kirkman triple system of order. Nonconstant arithmetic progression of length; Identification of the arithmetic triple order with common elements; Characterization of the Kirkman triple systems.