midpoint

The midpoint of a line segment AB is the point M that lies on AB and divides the segment into two congruent parts, so that AM = MB. In other words, A, M, and B are collinear and M is exactly halfway between A and B.

In coordinate geometry, if A has coordinates (x1, y1) and B has coordinates (x2, y2), the midpoint M has coordinates ((x1 + x2)/2, (y1 + y2)/2). This generalizes to higher dimensions by averaging corresponding coordinates: in n-dimensional space, M = ((x1 + x2)/2, (y1 + y2)/2, ..., (xn + xn)/2).

Key properties of the midpoint include its uniqueness and its status as the point on AB equidistant from A and B. In vector terms, the midpoint can be written as M = (a + b)/2, where a and b are the position vectors of A and B. The midpoint also represents the result of linear interpolation between A and B with parameter t = 1/2.

In triangle geometry, the segment joining the midpoints of two sides is called a midsegment; it is parallel to the third side and half as long. The concept extends to practical tasks such as dividing a segment into two equal parts, averaging coordinates in graphics and data analysis, and serving as a simple example of a convex combination.

Example: for A = (2, −3) and B = (8, 7), the midpoint is M = ((2+8)/2, (−3+7)/2) = (5, 2).