## When Less is More: Visualizing Basic InequalitiesInequalities permeate mathematics, from the Elements of Euclid to operations research and financial mathematics. Yet too often the emphasis is on things equal to one another rather than unequal. While equalities and identities are without doubt important, they don't possess the richness and variety that one finds with inequalities. The objective of this book is to illustrate how use of visualization can be a powerful tool for better understanding some basic mathematical inequalities. Drawing pictures is a well-known method for problem solving, and we would like to convince you that the same is true when working with inequalities. We show how to produce figures in a systematic way for the illustration of inequalities; and open new avenues to creative ways of thinking and teaching. In addition, a geometric argument can not only show two things unequal, but also help the observer see just how unequal they are. |

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### Contents

Representing positive numbers as lengths of segments | 1 |

Representing positive numbers as areas or volumes | 19 |

Inequalities and the existence of triangles | 43 |

Using incircles and circumcircles | 55 |

Using reflections | 73 |

Employing nonisometric transformations | 93 |

### Common terms and phrases

acute triangle algebraic Alsina altitudes AM-GM inequality angle bisectors Application Applying the AM-GM arithmetic mean Bernoulli's inequality Cauchy-Schwarz inequality Challenge Chapter circle circumcenter circumcircle circumradius concave construct convex cos2 Create a visual cyclic quadrilateral denote the area denote the lengths diagonals equality establish Euler's triangle inequality example form a triangle function graph gray triangle hence Heron's formula illustrated in Figure implies inequality is equivalent inradius isoperimetric theorem law of cosines Lemma line segment Mathematical mediant n-gon Nelsen nonnegative Padoa's inequality parallelogram perpendicular polygon positive numbers problem prove radius real numbers rectangle right triangle Ross Honsberger Schur's inequality secant line semiperimeter shown in Figure side lengths similarly slope subadditive superellipse tangent line trapezoid triangle ABC triangle given triangle inequality triangle is equilateral triangle with sides vectors vertex vertices visual proof Weitzenbock