Betweencenters

Betweencenters are a conceptual family of triangle centers defined by taking points that lie along the line segment joining two fixed reference centers of a triangle. The idea is to interpolate between two standard centers, such as the incenter and the circumcenter, the centroid and the circumcenter, or other pairs, to generate a continuum of points inside or near the triangle.

Formally, let ABC be a triangle and let X and Y be two chosen centers associated with ABC. For a parameter t in the interval [0,1], the betweencenter B(t) is the point on the line XY that divides the segment from X to Y in the ratio t:(1−t). In vector form, B(t) can be written as B(t) = (1−t)X + tY, and in barycentric coordinates the construction corresponds to taking a convex combination of the coordinates of X and Y. When t = 0, B(t) coincides with X; when t = 1, it coincides with Y.

Properties and variations of betweencenters depend on the chosen pair of reference centers. The construction is affine-invariant, so points behave predictably under similarity and affine transformations. The concept yields a one-parameter family of points, enabling exploration of how classical centers relate along a fixed line. Variants may allow t outside [0,1] to extend beyond the segment XY or use different reference centers to examine other alignments.

Note that betweencenters are not a standardized, universally defined class in all literature; definitions and preferred pairs of centers can vary across sources. The term is often used descriptively to discuss linear relationships between known centers.