Kolmiogeometria

Kolmiogeometria is a branch of geometry that studies triangles and triangle-based figures, focusing on their shapes, sizes, and the relationships that remain invariant under similarity and Euclidean motions. It explores both intrinsic properties of a triangle and the way triangles interact within larger configurations.

Core objects and concepts include the different types of triangles (acute, obtuse, right), the incircle and excircles, and the circumcircle. Central points such as the incenter, circumcenter, centroid, and orthocenter organize much of the theory. Important lines and circles include the Euler line, which passes through the circumcenter, centroid, and orthocenter, and the nine-point circle, which passes through the midpoints of the sides and the feet of the altitudes; its center lies on the Euler line and its radius is half the circumradius. The centroid divides each median in a 2:1 ratio. Classical relations involve the circumradius R, the inradius r, and distances such as OI, where O is the circumcenter and I the incenter, with the formula OI^2 = R(R − 2r).

Tools and methods combine synthetic and analytic approaches. Theorems like Ceva and Menelaus provide criteria for concurrency and collinearity in triangle configurations, while coordinate methods such as barycentric and trilinear coordinates allow systematic investigations of triangle properties and centers.

History and scope: kolmiogeometria has roots in classical Greek geometry and was developed through the 18th–19th centuries with advances in synthetic methods and coordinate techniques. It underpins applications in computer graphics, mesh generation, architectural design, and surveying, where precise triangle relationships are essential.