síðmiðja

Síðmiðja is an Icelandic geometric term that translates to “side midpoint” and denotes the midpoint of a side of a polygon, most commonly a triangle. It is the point on a given side that is equidistant from the side’s endpoints.

In a triangle, the three side midpoints are connected to form the medial triangle. This medial triangle is similar to the original triangle with a linear scale factor of 1/2, and its sides are each parallel to the corresponding sides of the original triangle. A defining property is that the segment joining the midpoints of two sides is parallel to the third side and has length equal to half that side.

Coordinates offer a simple computation: for a segment AB with endpoints A(x1, y1) and B(x2, y2), the side midpoint is M((x1 + x2)/2, (y1 + y2)/2). For polygons with more than three sides, the concept extends to the midpoints of individual sides, and these points can be used in various constructions and calculations, such as determining centroids or performing geometric transformations.

Construction methods include the standard compass-and-straightedge approach to locate the midpoint along a given side, or using coordinate geometry in a formal setting. The concept of side midpoints underpins several classical results in Euclidean geometry, particularly those involving medians, centroids, and the medial polygon.

See also: midpoint, median (geometry), centroid, medial triangle, polygon.