Geometric Construction (Problem and Ruler-and-Compass)

Greek geometry did not merely argue that figures could exist; it insisted you actually produce them using only the operations the postulates allow—drawing a line between two points and sweeping a circle with a compass. From this discipline came the deep division between a "problem" (a task to construct something) and a "theorem" (a truth to be demonstrated). It also bequeathed three famous puzzles—doubling the cube, trisecting an angle, and squaring the circle—that resisted ruler-and-compass solution for over two thousand years, until modern algebra finally proved them impossible.