Geometric Inequalities

Gangsong Leng, Yongming Liu

• Publisher: WSPC
• Year: 2015
• eBook ISBN: 9789814696500
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

In China, lots of excellent maths students take an active interest in various maths contests and the best six senior high school students will be selected to form the IMO National Team to compete in the International Mathematical Olympiad. In the past ten years China's IMO Team has achieved outstanding results — they won the first place almost every year.

The author is one of the coaches of China's IMO National Team, whose students have won many gold medals many times in IMO.

This book is part of the Mathematical Olympiad Series which discusses several aspects related to maths contests, such as algebra, number theory, combinatorics, graph theory and geometry. The book elaborates on Geometric Inequality problems such as inequality for the inscribed quadrilateral, the area inequality for special polygons, linear geometric inequalities, etc.

Request Inspection Copy

In China, lots of excellent maths students take an active interest in various maths contests and the best six senior high school students will be selected to form the IMO National Team to compete in the International Mathematical Olympiad. In the past ten years China's IMO Team has achieved outstanding results — they won the first place almost every year.

The author is one of the coaches of China's IMO National Team, whose students have won many gold medals many times in IMO.

This book is part of the Mathematical Olympiad Series which discusses several aspects related to maths contests, such as algebra, number theory, combinatorics, graph theory and geometry. The book elaborates on Geometric Inequality problems such as inequality for the inscribed quadrilateral, the area inequality for special polygons, linear geometric inequalities, etc.

Request Inspection Copy

Readership: Senior high school students engaged in math contests, math teachers, undergraduates of math major and math enthusiasts.
Key Features:

• China has performed outstandingly in IMO and the book gathers from the tutorial experience of many excellent teachers
• The author is one of the leading experts in aspects of maths contests in China as the coach of China's IMO National Team
• The Chinese version of the book has been very popular

The comparison of lengths is more basic than comparison of other geometric quantities (such as angles, areas and volumes). A geometric inequality that involves only the lengths is called a distance inequality.

Some simple axioms and theorems on inequalities in Euclidean geometry are usually the starting point to solve problems of distance inequality, in which most frequently used tools are:

Proposition 1. The shortest line connecting point A with point B is the segment AB.

The direct corollary of Proposition 1 is

Proposition 2 (Triangle Inequality). For arbitrary three points A, B and C, we have AB

AC + CB, the equality holds if and only if C is on the segment AB.

Remark. In this book, to simplify notations, any symbol of geometric object also denotes its quantity according to the context.

Proposition 2 has the following often used consequences.

Proposition 3. In a triangle, the longer side has the larger opposite angle. And the larger angle has the longer opposite side.

Proposition 4. The median of a triangle on a side is shorter than the half-sum of the other two sides.

Proposition 5. If a convex polygon is within another one, then the outside convex polygon’s perimeter is larger.

Proposition 6. Any segment in a convex polygon is either less than the longest side or the longest diagonal of the convex polygon.

Firstly, we give an example.

Example 1. Let a, b and c be sides of Δ ABC. Prove that

Proof. By the triangle inequality a < b + c, yields

Similarly,

Adding up the above three inequalities leads to Inequality (1).

Example 2. Let AB be the longest side of Δ ABC, an...

Table of contents

• Cover
• Halftitle
• Title Page
• Copyright
• Contents
• Preface
• Chapter 1 The method of segment replacement for distance inequalities
• Chapter 2 Ptolemy’s inequality and its application
• Chapter 3 Inequality for the inscribed quadrilateral
• Chapter 4 The area inequality for special polygons
• Chapter 5 Linear geometric inequalities
• Chapter 6 Algebraic methods
• Chapter 7 Isoperimetric and extremal value problem
• Chapter 8 Embed inequality and inequality for moment of inertia
• Chapter 9 Locus problem of Tsintsifas’s inequality
• Chapter 10 Shum’s minimal circle problem
• Chapter 11 Inequalities for tetrahedron
• Answers and hints to selected exercises