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alsina c nelsen r cornucopia quadrilaterals 2020

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Textbook in PDF format A Cornucopia of Quadrilaterals collects and organizes hundreds of beautiful and surprising results about four-sided figures—for example, that the midpoints of the sides of any quadrilateral are the vertices of a parallelogram, or that in a convex quadrilateral (not a parallelogram) the line through the midpoints of the diagonals (the Newton line) is equidistant from opposite vertices, or that, if your quadrilateral has an inscribed circle, its center lies on the Newton line. There are results dating back to Euclid: the side-lengths of a pentagon, a hexagon, and a decagon inscribed in a circle can be assembled into a right triangle (the proof uses a quadrilateral and circumscribing circle); and results dating to Erdos: from any point in a triangle the sum of the distances to the vertices is at least twice as large as the sum of the distances to the sides. The book is suitable for serious study, but it equally rewards the reader who dips in randomly. It contains hundreds of challenging four-sided problems. Instructors of number theory, combinatorics, analysis, and geometry will find examples and problems to enrich their courses. The authors have carefully and skillfully organized the presentation into a variety of themes so the chapters flow seamlessly in a coherent narrative journey through the landscape of quadrilaterals. The authors' exposition is beautifully clear and compelling and is accessible to anyone with a high school background in geometry. Preface Simple Quadrilaterals Introduction The Varignon parallelogram The quadrilateral law 1Diagonal midpoints, the Newton line, and Anne's theorem Area formulas and inequalities Van Aubel's theorem Equilic quadrilaterals Challenges Quadrilaterals and Their Circles Introduction Cyclic quadrilaterals The diagonals of a cyclic quadrilateral Brahmagupta' s formula Maltitudes and the anticenter of a cyclic quadrilateral Tangential quadrilaterals and Newton's theorem Bicentric quadrilaterals Extangential quadrilaterals and Urquhart's theorem Challenges Diagonals of Quadrilaterals Introduction Orthodiagonal quadrilaterals Equidiagonal quadrilaterals Kites Midsquare quadrilaterals Summary Challenges Properties of Trapezoids Introduction Pythagorean-like theorems for trapezoids Trapezoid area and the bimedians Trapezoid diagonals Isosceles trapezoids Trilateral trapezoids Right trapezoids Challenges Applications of Trapezoids Trapezoidal means Trapezoids and the Hermite-Hadamard inequality Trapezoidal reptiles and infinite series Fibonacci trapezoids Right trapezoids and the Erdos-Mordell inequality Challenges Garfield Trapezoids and Rectangles Introduction Inequalities for means Diophantus of Alexandria's sum of squares identity and the Cauchy-Schwarz inequality Trigonometric identities Arctangent identities Challenges Parallelograms Introduction Some basic parallelogram theorems Papp us' s area theorem Bhaskara and parallelograms The area of a parallelogram as a determinant Parallelograms in space The mediant property and Simpson's paradox Challenges Rectangles Introduction Characterizations Reciprocal rectangles Golden rectangles Silver and other metallic rectangles Fibonacci rectangles Challenges Squares Introduction Characterizations The bride's chair and the Vecten configuration Inscribing squares in triangles Squared squares and related objects Pythagorean triples Challenges Special Quadrilaterals Introduction Concave quadrilaterals Complex quadrilaterals Skew quadrilaterals Saccheri and Lambert quadrilaterals Challenges Quadrilateral Numbers Introduction Square and oblong numbers Square sums of two consecutive square numbers Trapezoidal and polite numbers Challenges Solutions to the Challenges Appendix: A Quadrilateral Glossary Credits and Permissions Bibliography Index

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Alsina C., Nelsen R. A Cornucopia of Quadrilaterals 2020.pdf 48.8 MB

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alsina c nelsen r cornucopia quadrilaterals 2020

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