Author’s Introduction
Zhong Jiucheng (Ya’an, Sichuan, China) Geometry has long been a cornerstone of human intellectual pursuit, offering a lens through which we can understand the world’s structure and beauty. From the ancient Greeks to modern mathematicians, the study of shapes, angles, and proportions has inspired countless breakthroughs in science, art, and engineering. This monograph is a humble contribution to this timeless field, exploring some of the most intriguing and challenging problems in classical geometry.
The journey of this work begins with the fundamental concept of the angle bisector, a line that divides an angle into two equal parts. This simple yet profound idea serves as the foundation for many geometric constructions and proofs. Here, we delve into the principle that any angle has a bisector that can be constructed using a ruler, demonstrating the elegance and universality of this geometric tool.
Building on this, the monograph explores the ruler’s ability to construct a line segment equal to one-third of the chord length of any angle. This seemingly straightforward task reveals the depth of geometric reasoning required to achieve precision and accuracy in construction.
The work then ventures into the realm of ancient mathematical challenges, addressing the classic problem of squaring the circle. Using the Pythagorean Theorem, we demonstrate how a square with an area equal to that of a given circle can be constructed—a problem that has fascinated mathematicians for centuries.
Another cornerstone of this monograph is the exploration of the doubling of the cube, a problem rooted in Greek antiquity. By using a given cube edge, we show how to construct an edge equal to twice the original cube’s edge length, offering insights into the interplay between algebra and geometry.
The study also extends to the construction of polygons, particularly those with a positive nonagon or a number of sides that is a power of three. These constructions highlight the versatility of the ruler and the enduring appeal of geometric symmetry.
Throughout this monograph, the process of solving these difficult problems is meticulously recorded. Each step reflects the challenges, insights, and triumphs encountered along the way, offering readers a glimpse into the creative and analytical process that underpins geometric discovery.
This work would not have been possible without the support and inspiration of many individuals. I extend my heartfelt gratitude to my friends who have guided and encouraged me throughout this journey. Their insights and feedback have been invaluable in shaping this monograph.
