Triangleiden

Triangleiden is a geometric construction named for its triangular symmetry. Given a fixed reference triangle ABC in the plane, a triangleiden is defined as a triangle T that is inscribed in ABC with its three vertices each lying on a different side of ABC and such that T is similar to ABC. The term is used in some puzzle and recreational geometry contexts to study families of inscribed similar triangles and their scaling behavior.

For a chosen similarity ratio r with 0 < r < 1, there is typically a unique inscribed triangle T that is similar to ABC and whose vertices lie on the three sides of ABC in the prescribed order. As r varies, the corresponding triangleidens form a one-parameter family, tracing configurations from the reference triangle toward the medial position when r equals 1/2. This family provides a simple model for examining how a fixed triangle can appear in many scaled forms within its own boundaries while preserving shape.

Properties and relations: Because all triangleidens are similar to ABC, they share the same set of angles as ABC. The area of T scales with r^2 relative to the area of ABC, and the linear dimensions of T scale with r. The construction is closely related to classical notions of similarity and homothety, and it offers a convenient framework for exploring inscribed figures and scale transformations within a single triangle.

See also: Inscribed triangle, Medial triangle, Homothety, Similarity.