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Midpoint

In , the midpoint of a is the point that divides the segment into two parts of equal length, making it from both endpoints. This point serves as the center of the segment and is also its , representing the balance point if the segment were considered a one-dimensional distribution. In coordinate , the coordinates of the midpoint between two points (x_1, y_1) and (x_2, y_2) are calculated using the : \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). This arises from averaging the respective coordinates, effectively finding the in each dimension. It applies to segments in the and can extend to higher dimensions, such as , by including a z-coordinate in the . The concept of the midpoint is fundamental in various geometric constructions and proofs, including the midpoint theorem, which relates the midpoints of a triangle's sides to a parallel to the third side and half as long. Beyond pure geometry, midpoints appear in for determining coordinates of medians in triangles and in physics for calculating centers of mass in simple systems. In statistics, the term midpoint can refer to the average value of an interval in , though this usage stems from the same averaging principle.

Definition and Formulas

Basic Definition

In , the midpoint of a AB is defined as the unique point M on the segment such that the AM equals the MB, with M positioned between A and B. This division creates two congruent subsegments, ensuring balance and equality along the line. Conceptually, the midpoint represents a 1:1 ratio division of the segment, where the position of M balances the endpoints A and B. In terms, this corresponds to the position of M being the arithmetic of the position vectors of A and B, emphasizing the symmetric distribution of the segment's extent. The notion of the midpoint traces its origins to in Euclid's Elements, composed around 300 BCE, particularly in Book I, Proposition 10, which addresses the of a finite straight line to locate its midpoint. Visually, the midpoint on a straight line illustrates perfect symmetry, appearing as the central point where the folds onto itself without overlap, highlighting the inherent equilibrium in linear division.

Coordinate Formula

In coordinate geometry, the midpoint M of a joining two points A(x_1, y_1) and B(x_2, y_2) in the plane is given by the M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). This arises from the principle of averaging the coordinates, as the midpoint represents the of the endpoints' positions along each , ensuring it lies exactly halfway between them. To illustrate, consider points A(1, 2) and B(3, 4). The midpoint is M = \left( \frac{1 + 3}{2}, \frac{2 + 4}{2} \right) = (2, 3). This calculation confirms the point's position at the . The formula's validity can be verified using the distance formula, which states the distance between two points (x_a, y_a) and (x_b, y_b) is \sqrt{(x_b - x_a)^2 + (y_b - y_a)^2}. Let M be the proposed midpoint. The distance AM simplifies to AM = \sqrt{\left( \frac{x_2 - x_1}{2} \right)^2 + \left( \frac{y_2 - y_1}{2} \right)^2} = \frac{1}{2} \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, and similarly for BM, yielding AM = BM, thus confirming equal distances from M to the endpoints. This approach extends to , where for points A(x_1, y_1, z_1) and B(x_2, y_2, z_2), the midpoint is M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right), by averaging the coordinates across all three axes in an analogous manner.

Vector Formula

In , the position of the midpoint \vec{M} of two points with position vectors \vec{A} and \vec{B} is given by \vec{M} = \frac{\vec{A} + \vec{B}}{2}. This formula holds in any over a where division by 2 is defined, such as the . This expression derives from the in vector terms, which describes the position of a point dividing the segment joining \vec{A} and \vec{B} in the m:n as \frac{m \vec{B} + n \vec{A}}{m + n}. For the midpoint, the ratio is 1:1, reducing to the average. More fundamentally, it arises as a special case of an affine combination, where \vec{M} = t \vec{A} + (1 - t) \vec{B} with t = \frac{1}{2}, ensuring the coefficients sum to 1 and preserving affine structure. Key properties include the midpoint serving as the centroid of the two points under equal weighting, equivalent to the center of mass for two equal masses at \vec{A} and \vec{B}. Additionally, the midpoint is invariant under translation: if both points are shifted by a vector \vec{T}, the new midpoint is \frac{(\vec{A} + \vec{T}) + (\vec{B} + \vec{T})}{2} = \vec{M} + \vec{T}, maintaining its relative position. In , the midpoint between vectors \vec{u} and \vec{v} is \frac{\vec{u} + \vec{v}}{2}, which applies to calculations; for example, if \vec{u} and \vec{v} are vectors from a reference , this represents the to the midpoint, useful in analyzing symmetric motion or balancing forces in . This vector approach generalizes the coordinate-based formula to arbitrary dimensions without relying on a specific basis.

Construction Techniques

Compass and Straightedge Method

The and method is a classical technique in for constructing the midpoint of a given , relying solely on these two tools to ensure precision without measurement. The allows for drawing circles of specified radii, while the enables the creation of straight lines connecting points or extending lines. This approach is foundational to geometric constructions, as it adheres to the postulates outlined in . Historically, the method originates from 's Elements, specifically Book I, Proposition 10, where it is presented as a way to bisect a given finite straight line. Euclid demonstrates that by constructing the bisector of the segment, the point of intersection with the original line serves as the midpoint, equidistant from both endpoints. This proposition builds on earlier postulates regarding the use of the and , establishing a rigorous procedure that has influenced geometric practice for over two millennia. To construct the midpoint of a AB using this method, begin by drawing the segment AB with the . Place the point at A and adjust the to the length of AB, then draw a (or ) centered at A that passes through B; repeat the process with the centered at B, drawing another of the same that passes through A. These two circles will intersect at two points, say C and D, which are from A and B. Use the to draw the line connecting C and D, forming the bisector of AB. The point where this bisector intersects AB is the midpoint M. This works because any point on the bisector is from A and B by the properties of circles. The method assumes the line segment is straight and finite, as defined in , and cannot directly apply to curved or infinite lines without additional constructions. In practice, the accuracy of the resulting midpoint depends on the precision of the and used, as small errors in drawing circles or lines can accumulate, leading to deviations from the true midpoint.

Coordinate Geometry Method

The coordinate geometry method for determining the midpoint of a line segment relies on algebraic within a , where endpoints are represented as ordered pairs. To apply this method, assign coordinates to the two endpoints of the segment, denoted as (x_1, y_1) and (x_2, y_2). The midpoint M is then found by averaging these coordinates using the formula: M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). This averaging process yields the precise location of the midpoint, which can be plotted on a graph or incorporated into subsequent geometric calculations. The procedure begins by identifying the endpoint coordinates, followed by separately adding and dividing the x-values and y-values by 2 to obtain the midpoint pair. Once computed, the result can be visualized by marking the point on the coordinate plane or used directly in analytical tasks. For instance, for a line segment connecting (0,0) and (4,6), the midpoint is calculated as \left( \frac{0 + 4}{2}, \frac{0 + 6}{2} \right) = (2, 3). To confirm its position, the Euclidean distance from (0,0) to (2,3) equals \sqrt{(2-0)^2 + (3-0)^2} = \sqrt{13}, matching the distance from (2,3) to (4,6) at \sqrt{(4-2)^2 + (6-3)^2} = \sqrt{13}, verifying the midpoint divides the segment equally. This approach provides high precision in numerical results, avoiding approximations inherent in graphical constructions, and proves advantageous in applications requiring exact computations, such as positioning elements in or in . In programming environments, the is straightforward to implement; for example, supports creating a simple function to compute the midpoint by applying the averaging to input coordinates, facilitating in geometric software.

Properties in Euclidean Geometry

In Line Segments

In Euclidean geometry, the midpoint of a line segment is defined as the point that divides the segment into two congruent parts of equal length. This property ensures that the distances from the midpoint to each endpoint are identical, providing a balanced division fundamental to many geometric constructions and proofs. A key implication of this property concerns the locus of midpoints formed by segments connecting two parallel lines. Specifically, the set of all midpoints of line segments with one endpoint on each of two parallel lines forms a straight line parallel to both and positioned midway between them. This locus arises because the y-coordinate (in a aligned with the lines) of each midpoint is consistently the average of the fixed distances of the parallel lines, while the x-coordinates vary freely along the direction of the lines. Another significant property involves connecting midpoints in polygonal figures. For instance, the midpoints of the sides of any , when joined in order, form a known as the Varignon parallelogram. This result highlights how midpoints preserve parallelism and relationships even in non-parallelogram quadrilaterals. In the context of transversals, the segment joining the midpoints of two sides intersected by a transversal exhibits parallelism and proportional length. A brief extension to triangular setups shows that such a segment is parallel to the third side and half its length, underscoring the midpoint's role in similarity and proportion. Midpoints find practical application in partitioning problems, where they enable equal division of paths or segments, such as determining balanced positions in or coordinate systems. For example, in dividing a route equally between two points, the midpoint serves as the central reference, facilitating symmetric allocations in geometric modeling.

In Triangles

In a triangle, the line segment connecting the midpoints of two sides is known as a midsegment. The midsegment states that this segment is to the third side of the triangle and half as long as that side. To prove the midsegment theorem using similar triangles, consider triangle ABC with midpoints D and E on sides AB and AC, respectively, so DE is the midsegment. Draw line DE parallel to BC by construction or assumption in the proof setup. Triangles ADE and ABC share angle A, and corresponding angles at D and B, as well as at E and C, are equal due to parallel lines and transversals AB and AC. Thus, \triangle ADE \sim \triangle ABC by AA similarity, implying DE/BC = AD/AB = 1/2, so DE = BC/2 and DE \parallel BC. A median of a triangle is the line segment from a vertex to the midpoint of the opposite side. The three medians of any triangle are concurrent, intersecting at a single point called the centroid, which divides each median in a 2:1 ratio, with the longer segment closer to the vertex. To establish concurrency, one approach uses vector geometry: assign position vectors to vertices A, B, C; the midpoint of BC is (B + C)/2, so the median from A is parameterized as A + t((B + C)/2 - A). Setting the intersection of two medians and solving shows the third passes through the same point at t = 2/3. Alternatively, affine transformations preserve midpoints and ratios, mapping any triangle to an equilateral one where medians concur visibly, implying general concurrency. Apollonius's theorem relates the sides and medians of a : for m_a from A to the midpoint of side a = BC, the equation is b^2 + c^2 = 2m_a^2 + \frac{a^2}{2}, where b = AC and c = AB. This follows as a special case of Stewart's theorem applied to the median, where the cevian divides the base into equal segments of a/2. Rearranging yields the median length m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2}. In an equilateral triangle with side length s, the medians coincide with the altitudes, angle bisectors, and perpendicular bisectors, each with length \frac{\sqrt{3}}{2}s, and the centroid is also the center of symmetry.

In Quadrilaterals and Polygons

In quadrilaterals, a fundamental property involving midpoints is given by Varignon's theorem, which states that the figure formed by joining the midpoints of the sides of any quadrilateral is a parallelogram, known as the Varignon parallelogram. This holds for both convex and concave quadrilaterals, and the Varignon parallelogram has an area equal to half that of the original quadrilateral. A proof of Varignon's theorem can be obtained using vector geometry by considering the position vectors of the quadrilateral's vertices, denoted as \vec{A}, \vec{B}, \vec{C}, and \vec{D}. The midpoints have position vectors \vec{M} = \frac{\vec{A} + \vec{B}}{2}, \vec{N} = \frac{\vec{B} + \vec{C}}{2}, \vec{P} = \frac{\vec{C} + \vec{D}}{2}, and \vec{Q} = \frac{\vec{D} + \vec{A}}{2}. The vector \vec{MN} = \vec{N} - \vec{M} = \frac{\vec{C} - \vec{A}}{2} equals \vec{QP} = \vec{P} - \vec{Q} = \frac{\vec{C} - \vec{A}}{2}, showing that opposite sides are equal and parallel; similarly, \vec{NP} = \frac{\vec{D} - \vec{B}}{2} = \vec{QM}. Thus, MNOP is a parallelogram. Specific cases illustrate further properties of the Varignon . For a , which has equal diagonals, the Varignon parallelogram is a , as the sides of the resulting figure are equal in length due to the of the diagonals. Extending to general , the midpoint polygon is formed by connecting the midpoints of the original polygon's sides, resulting in a new polygon with the same number of sides. For a regular n-gon, the midpoint polygon is also regular, scaled by a factor related to the original's geometry and rotated by \pi/n radians relative to the original. For example, in a regular , the midpoint pentagon is regular with side lengths equal to half the diagonal length of the original pentagon. In polygons, midpoints also relate to the . The of the vertices, given by the position \frac{1}{n} \sum_{i=1}^n \vec{V_i}, coincides with the of the midpoints of the sides, since the of the midpoints \frac{1}{n} \sum_{i=1}^n \frac{\vec{V_i} + \vec{V_{i+1}}}{2} = \frac{1}{n} \sum_{i=1}^n \vec{V_i}. This provides a to locate the using side midpoints.

In Conic Sections

In conic sections, the midpoint of a plays a significant role in understanding geometric properties, particularly in , ellipses, and hyperbolas. For a , the from to any bisects the chord, meaning the midpoint lies on this line. This follows from the of the , where the two segments from to the endpoints form congruent right triangles. Specifically, if the center is at O and the is AB with midpoint M, then OM is to AB. Additionally, when the is a , its midpoint coincides with of , as the passes through by definition. In an ellipse, the locus of the midpoints of a family of parallel chords forms a straight line known as a , which passes through the center of the ellipse. For chords parallel to one of the principal axes, such as the minor axis, the midpoints lie along the major axis, reflecting the ellipse's . Regarding focal properties, the midpoints of focal chords—chords passing through one of the foci—trace a conic section similar to the original but scaled by a factor involving the e. For a \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, the midpoints of passing through a have a locus that is another . This property arises from the representation and the condition that the chord passes through the focus. In general conic sections, the midpoint of a between two points can be expressed using equations. For an \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, parameterized as (x_1, y_1) = (a \cos \theta, b \sin \theta) and (x_2, y_2) = (a \cos \phi, b \sin \phi), the midpoint is \left( \frac{a}{2} (\cos \theta + \cos \phi), \frac{b}{2} (\sin \theta + \sin \phi) \right). This formulation highlights how midpoints inherit the conic's scaling along the axes. Similar expressions apply to hyperbolas using , such as \cosh and \sinh, to describe midpoints. As an illustrative example in the unit circle x^2 + y^2 = 1, consider the between points (\cos \alpha, \sin \alpha) and (\cos \beta, \sin \beta); its midpoint is \left( \frac{\cos \alpha + \cos \beta}{2}, \frac{\sin \alpha + \sin \beta}{2} \right), which lies inside the unless the is a . In contrast, the midpoint of the corresponding (the point halfway along the ) is at \frac{\alpha + \beta}{2}, given by (\cos \frac{\alpha + \beta}{2}, \sin \frac{\alpha + \beta}{2}), emphasizing the distinction between linear midpoints and angular midpoints.

Advanced Generalizations

In Higher Dimensions

In n-dimensional \mathbb{R}^n, the midpoint \mathbf{M} of two points \mathbf{A} = (a_1, \dots, a_n) and \mathbf{B} = (b_1, \dots, b_n) is defined by the vector formula \mathbf{M} = \frac{\mathbf{A} + \mathbf{B}}{2}, or componentwise as M_i = \frac{a_i + b_i}{2} for each i = 1, \dots, n. This generalizes the concept from lower dimensions and relies on the structure of \mathbb{R}^n. The operation is affine invariant, meaning it yields the same result regardless of the chosen . A key geometric in higher dimensions involves hyperspheres, the n-dimensional analogs of circles and . The n- is the set of points in \mathbb{R}^{n+1} at a fixed distance (radius r) from a \mathbf{C}. For any diameter—a line segment passing through the with endpoints on the hypersphere—the midpoint of that segment coincides exactly with \mathbf{C}, as the endpoints are symmetric about \mathbf{C} in the Euclidean metric. This holds because if the endpoints are \mathbf{C} - r\mathbf{u} and \mathbf{C} + r\mathbf{u} for a unit vector \mathbf{u}, their average is \mathbf{C}. Applications of midpoints extend to physics and . In the 4-dimensional of , modeled as (a pseudo-Euclidean structure approximating for certain calculations), midpoints along worldlines—the paths traced by particles through —represent average event positions and times, aiding analysis of observer frames. For instance, in spacetime diagrams, an observer positioned at the midpoint of a platform detects simultaneous events in their , illustrating the . In , midpoints facilitate n-dimensional and ; they compute intermediate positions for smooth animations, interpolations in vector-based rendering, and coordinate transformations in higher-dimensional data plots, ensuring consistency across dimensions. Midpoints exhibit important algebraic and geometric properties in \mathbb{R}^n. Convexity is preserved under the midpoint operation: if \mathbf{A} and \mathbf{B} belong to a K \subseteq \mathbb{R}^n, then \frac{\mathbf{A} + \mathbf{B}}{2} \in K, since convex sets are closed under affine combinations with non-negative weights summing to 1. Additionally, midpoints of points within an affine subspace remain in that subspace, as the midpoint is the affine combination with equal weights $1/2, and affine subspaces are precisely those closed under such operations. The collection of all such midpoints spans the affine hull, reinforcing the role of midpoints in generating affine structures from point sets.

In Metric Spaces

In a metric space (X, d), a point m \in X is defined as a midpoint of two points a, b \in X if it satisfies d(a, m) = d(m, b) = \frac{1}{2} d(a, b). This generalizes the Euclidean notion by relying solely on the distance function, without assuming vector structure or straight-line segments. Unlike in Euclidean spaces, midpoints in general metric spaces may not exist, may not be unique, or may lie off any intuitive "straight" path between a and b. The existence of midpoints is closely tied to the geometry of the space. In a , where every pair of points is joined by at least one isometric embedding of an (a ), there is always at least one midpoint for any a, b \in X. Conversely, in length spaces—complete s where the equals the infimum of lengths of curves connecting points—midpoints can be approximated arbitrarily closely, and completeness ensures the existence of actual midpoints along minimizing paths. However, uniqueness fails in many cases; for instance, in discrete metric spaces like finite graphs with integer edge lengths, if d(a, b) is odd, no midpoint exists at all, as distances cannot halve evenly. A classic example of non-uniqueness occurs in taxicab geometry, or the L^1 metric on \mathbb{R}^2, where d((x_1,y_1), (x_2,y_2)) = |x_1 - x_2| + |y_1 - y_2|. For points not aligned horizontally or vertically, such as (0,0) and (2,2), the set of midpoints forms the line segment connecting (2,0) and (0,2), all at distance 2 from each endpoint. In contrast, metric trees—geodesic spaces with unique paths between any two points—always admit a unique midpoint, lying exactly halfway along the isometric embedding of [0, d(a,b)] between a and b. These properties highlight how midpoints encode convexity and path-minimization in abstract settings, influencing applications in optimization and topology.

In Graph Theory

In graph theory, a midpoint between two vertices u and v in an undirected connected graph G is a vertex w that lies on at least one shortest u-v path and satisfies d_G(u, w) = d_G(w, v) = d_G(u, v)/2, where d_G denotes the shortest-path distance in G and d_G(u, v) is even. This concept extends the geometric midpoint to discrete structures, where shortest paths serve as geodesics, and such w exists only if the distance is even; otherwise, no vertex midpoint is possible, though edge midpoints could be considered in metric graphs with edge lengths. The midpoint set for a pair (u, v) is the collection of all such vertices w. For a G, the overall midpoint set can be defined as the union of midpoint sets over all pairs of vertices at maximum (the ), capturing vertices central to the graph's longest connections. This set relates to the graph's and : the r is the minimum (maximum from a to any other), the d is the maximum , and in general graphs, r ≤ d ≤ 2r. Every central (those with r) lies on some diametral (shortest of length d), and thus the is a of the midpoints of diametral paths when d is even. In trees, this relation is exact: the consists precisely of the midpoint(s) of any diametral , with a single central if d is even (r = d/2) or two adjacent central vertices if d is odd (r = (d+1)/2). To compute the midpoint set for a specific pair (u, v), perform breadth-first search (BFS) from u to compute all distances and identify d = d_G(u, v); if d is odd, the set is empty. If even, let k = d/2; run BFS from u to find all vertices at distance k from u, then run BFS from v to verify which of those are at distance k from v (automatically ensuring they lie on some shortest path since d_G(u, w) + d_G(w, v) = d). This requires two BFS traversals, running in O(|V| + |E|) time, where |V| is the number of vertices and |E| the number of edges. Applications of midpoints appear in network analysis, where the midpoint set helps identify balanced locations for facilities or hubs that equidistant from key nodes, minimizing maximum travel in transportation or communication networks. In social networks, midpoint centrality—measuring the fraction of pairs for which a vertex serves as a midpoint—highlights influential nodes bridging distant actors, similar to betweenness but focused on balanced positions; higher midpoint centrality indicates structural importance in global productive service networks. For example, in a path graph P_5 (vertices 1-2-3-4-5), the midpoint set between endpoints 1 and 5 (d=4) is {3}, the unique central vertex. In a cycle graph C_6 (vertices 1-2-3-4-5-6-1), the distance between 1 and 4 is 3 (odd), so the midpoint set is empty; between 1 and 5, d=2 (via 1-6-5), and the midpoint set is {6}. In C_8 (vertices 1-2-3-4-5-6-7-8-1), between 1 and 5 (d=4), the shortest paths are 1-2-3-4-5 and 1-8-7-6-5; the midpoint set is {3,7}.