midpoints

Midpoint is a point that divides a line segment into two congruent segments. In Euclidean geometry, the midpoint M of a segment AB is the point on AB such that AM = MB. The concept extends to line segments in any dimension, with the same equidistance property on the segment AB.

Coordinate form: If A(x1, y1) and B(x2, y2) then the midpoint M is ((x1 + x2)/2, (y1 + y2)/2). In 3D, M = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2). In vector terms, M is the average of the position vectors of A and B: M = (A + B) / 2.

Construction: With a compass and straightedge, one common method is to draw circles centered at A and B with equal radius r > AB/2. The circles intersect at two points; drawing the line through these intersection points yields the perpendicular bisector of AB. Its intersection with AB is the midpoint. This method relies on the symmetry of the circles about AB.

Properties: By definition, AM = MB, making M the center of symmetry for the segment AB. In triangles, the midpoint theorem states that the segment joining the midpoints of two sides is parallel to the third side and has half its length; this is a key tool in geometric proofs and constructions.

Applications: Midpoints are used to divide segments into equal parts, to locate centers in geometric constructions, and in fields such as computer graphics, surveying, and geographic information systems. They also provide a basis for more complex concepts like centroids in polygons and parameterizations in computational geometry.