What is a Corollary in Mathematics?

In mathematics, a corollary is a result that follows almost immediately and naturally from a previously proven theorem or proposition. Corollaries are typically direct consequences of a larger result, requiring minimal additional proof. They are often used to highlight interesting or useful implications that stem directly from a more fundamental result.

Key Points About Corollaries

1. Derived Result: Corollaries are usually straightforward conclusions that stem from a major result.
2. Minimal Proof: Because they follow directly from a theorem or proposition, corollaries need little to no additional proof beyond that of the original result.
3. Highlight Implications: Corollaries help underscore the broader implications of a theorem, often making it easier to apply or understand.

For example, after proving the Pythagorean Theorem, a corollary is that in a right triangle, the length of the altitude to the hypotenuse can be determined using the two legs. This corollary expands on the theorem by providing an additional, directly related result.