Katetuoton

Katetuoton is a term found in some mathematical education contexts to denote a specific dimensionless quantity associated with a right triangle. It is not part of the standard mathematical canon, but it appears in problem sets and explanations to compare the two legs of a right triangle.

For a right triangle with leg lengths a and b and hypotenuse c, the katetuoton K is defined as K = c^2/(ab). Since c^2 = a^2 + b^2, this is equivalently K = a/b + b/a, a scale-invariant expression in the leg lengths.

Properties include scale invariance: multiplying all sides by a common factor t leaves K unchanged. The quantity is bounded below by 2, with equality at a = b; as the leg lengths become highly unequal, K increases without bound.

Example: In a 3-4-5 triangle, K = 5^2/(3×4) = 25/12 ≈ 2.0833, which also equals 3/4 + 4/3.

Usage and notes: The katetuoton serves as a compact measure of balance between the legs and is sometimes employed to illustrate inequalities or to provoke algebraic manipulation in teaching. It has no widely adopted theorems beyond the simple identity K = a/b + b/a and the AM-GM bound.

Etymology: The name draws on kathetos, the Greek word for a cathetus (leg) of a right triangle, with a suffix forming a coined mathematical term. The concept is not widely standardized and remains largely a pedagogical construct.