corollaries

Corollaries are propositions that follow directly from a previously proven theorem, typically with little or no additional proof required. They express natural consequences or applications of the main result and are often presented immediately after the theorem to illustrate its power. The term comes from the Latin corollarium, meaning a garland or reward attached to a prize, used metaphorically to denote an attached consequence. In mathematical writing, a corollary is usually a direct, almost self-evident consequence of the theorem. Corollaries are distinguished from lemmas, which serve as intermediate steps to prove a theorem, and from propositions, which are interesting results but not as central as the main theorems.

Corollaries appear across disciplines that use formal proof, including geometry, analysis, algebra, and number theory. They help to highlight specific consequences and facilitate further reasoning by providing readily usable facts. For example, from the Pythagorean theorem, in a right triangle with legs a and b and hypotenuse c, the altitude to the hypotenuse has length h = ab/c, a direct corollary obtained via similar triangles. From the mean value theorem, if a function has derivative zero on an interval, the function must be constant. From the area formula A = πr^2, doubling the radius increases the area by a factor of four. Such corollaries illustrate how a single theorem can yield several concise, useful results.