midpunt

Midpunt, in geometry, is the point that divides a finite line segment into two congruent parts. For a segment with endpoints A and B, the midpunt M is the point on AB such that MA equals MB, or equivalently the distances from M to A and from M to B are equal. The concept is independent of the coordinate system and applies in two-dimensional and three-dimensional space. In many languages, including Dutch, midpunt is the standard term for this point.

In coordinates, if A has coordinates (x1, y1) and B has coordinates (x2, y2) in the plane, the midpunt M has coordinates ((x1+x2)/2, (y1+y2)/2). In three dimensions with z coordinates, M = ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2). In vector form, M can be written as M = (A + B) / 2. The distance AB equals twice the distance AM, i.e., AB = 2·AM = 2·MB.

Properties of the midpunt include that it lies on the segment AB and that AB is divided into two equal segments by M. The midpoint is also the intersection of AB with its perpendicular bisector; the perpendicular bisector is the locus of points equidistant from A and B. The midpunt is unique for each given segment.

Construction and applications: A standard compass-and-straightedge construction can locate the midpunt by drawing circles centered at A and B with radius AB; their intersections determine the perpendicular bisector, which crosses AB at the midpunt. In practice, midpunten are used in computer graphics, geographic information systems, surveying, and as a basic tool in solving a variety of geometry problems that require averaging positions or locating centers along a segment.