Kolmiovarianteoria

Kolmiovarianteoria is a theoretical framework in triangle geometry that investigates how triangles can vary under three coupled deformation modes. It treats three interdependent quantities—such as the side lengths a, b, c, or a combination of sides, angles, and an invariant like area or circumradius—and asks how these quantities change when one or more parameters are adjusted. The theory emphasizes the relationships among the three varying aspects rather than any single triangle in isolation.

A core concept is the tri-variant parameterization, which represents the triangle by a triple of deformation parameters (p, q, r) that control changes in shape, scale, and orientation. Kolmiovarianteoria seeks invariant relations that remain valid within prescribed families of deformations, mirroring classical results such as Ceva’s or Menelaus’ in a three-parameter setting and generalizing them for multi-quantity constraints.

Mathematically, the framework defines a deformation space in which coordinates encode triangle data and the parameters define admissible paths through that space. Differential equations or discrete update rules describe how the triangle evolves under a fixed variant rule. In special cases, the framework reduces to standard triangle geometry; for other choices it connects with non-Euclidean or discrete geometric contexts.

The theory remains largely theoretical and exploratory, with no standard catalog of results. It is used mainly to provide a unified language for discussing families of triangles across geometries and deformation regimes, and it has potential applications in advanced geometry education and in the modeling of systems with multi-quantity geometric constraints.