Mathematics abounds with unsolved problems, and among these, four classic geometric construction challenges have been deemed “unsolvable” for nearly two and a half millennia. This book devotes four chapters to presenting innovative proposals for their resolution. Additionally, two equally compelling topics—the construction of square roots for all numbers on the number line and the construction of regular polygons—are explored in dedicated sections. The remaining twelve chapters extend and extrapolate upon these themes. All constructions are within the framework of Euclidean geometry, with reverse construction methods examined wherever applicable. The involvement of irrational numbers in these problems is inevitable.