中点

Midpoint — Grokipedia Fact-checked by Grok 2 months ago Midpoint Ara Eve Leo Sal 1x In geometry , the midpoint of a line segment is the point that divides the segment into two parts of equal length, making it equidistant from both endpoints. [1] This point serves as the center of the segment and is also its centroid , representing the balance point if the segment were considered a one-dimensional mass distribution. [1] In coordinate geometry , the coordinates of the midpoint between two points ( x 1 , y 1 ) (x_1, y_1) ( x 1 ​ , y 1 ​ ) and ( x 2 , y 2 ) (x_2, y_2) ( x 2 ​ , y 2 ​ ) are calculated using the midpoint formula : ( x 1 + x 2 2 , y 1 + y 2 2 ) \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ( 2 x 1 ​ + x 2 ​ ​ , 2 y 1 ​ + y 2 ​ ​ ) . [2] This formula arises from averaging the respective coordinates, effectively finding the arithmetic mean in each dimension. [3] It applies to segments in the Euclidean plane and can extend to higher dimensions, such as three-dimensional space , by including a z-coordinate in the average . [4] The concept of the midpoint is fundamental in various geometric constructions and proofs, including the midpoint theorem and its converse, where the theorem states that the line segment joining the midpoints of two sides is parallel to the third side and half as long, and the converse states that if a line is drawn through the midpoint of one side of a triangle and is parallel to another side, then it bisects the third side. [5] Beyond pure geometry, midpoints appear in analytic geometry for determining coordinates of medians in triangles and in physics for calculating centers of mass in simple systems. [6] In statistics, the term midpoint can refer to the average value of an interval in grouped data , though this usage stems from the same averaging principle. [3] Definition and Formulas Basic Definition In geometry , the midpoint of a line segment AB is defined as the unique point M on the segment such that the length AM equals the length MB, with M positioned between A and B. This division creates two congruent subsegments, ensuring balance and equality along the line. [7] Conceptually, the midpoint represents a 1:1 ratio division of the segment, where the position of M balances the endpoints A and B. In vector terms, this corresponds to the position vector of M being the arithmetic average of the position vectors of A and B, emphasizing the symmetric distribution of the segment's extent. [8] The notion of the midpoint traces its origins to Euclidean geometry in Euclid's Elements , composed around 300 BCE, particularly in Book I, Proposition 10, which addresses the bisection of a finite straight line to locate its midpoint. [7] Visually, the midpoint on a straight line illustrates perfect symmetry, appearing as the central point where the segment folds onto itself without overlap, highlighting the inherent equilibrium in linear division. Coordinate Formula In coordinate geometry, the midpoint $ M $ of a line segment joining two points $ A(x_1, y_1) $ and $ B(x_2, y_2) $ in the plane is given by the formula M = ( x 1 + x 2 2 , y 1 + y 2 2 ) . M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). M = ( 2 x 1 ​ + x 2 ​ ​ , 2 y 1 ​ + y 2 ​ ​ ) . This formula arises from the principle of averaging the coordinates, as the midpoint represents the arithmetic mean of the endpoints' positions along each axis, ensuring it lies exactly halfway between them. [2] To illustrate, consider points $ A(1, 2) $ and $ B(3, 4) $. The midpoint is M = ( 1 + 3 2 , 2 + 4 2 ) = ( 2 , 3 ) . M = \left( \frac{1 + 3}{2}, \frac{2 + 4}{2} \right) = (2, 3). M = ( 2 1 + 3 ​ , 2 2 + 4 ​ ) = ( 2 , 3 ) . This calculation confirms the point's position at the average . [2] 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 A M = ( x 2 − x 1 2 ) 2 + ( y 2 − y 1 2 ) 2 = 1 2 ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 , 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}, A M = ( 2 x 2 ​ − x 1 ​ ​ ) 2 + ( 2 y 2 ​ − y 1 ​ ​ ) 2 ​ = 2 1 ​ ( 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. [9] This approach extends to three-dimensional space , where for points $ A(x_1, y_1, z_1) $ and $ B(x_2, y_2, z_2) $, the midpoint is M = ( x 1 + x 2 2 , y 1 + y 2 2 , z 1 + z 2 2 ) , M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right), M = ( 2 x 1 ​ + x 2 ​ ​ , 2 y 1 ​ + y 2 ​ ​ , 2 z 1 ​ + z 2 ​ ​ ) , by averaging the coordinates across all three axes in an analogous manner. [2] Vector Formula In vector geometry , the position vector of the midpoint $ \vec{M} $ of two points with position vectors $ \vec{A} $ and $ \vec{B} $ is given by M ⃗ = A ⃗ + B ⃗ 2 . \vec{M} = \frac{\vec{A} + \vec{B}}{2}. M = 2 A + B ​ . This formula holds in any vector space over a field where division by 2 is defined, such as the real numbers . [10] This expression derives from the section formula in vector terms, which describes the position vector of a point dividing the segment joining $ \vec{A} $ and $ \vec{B} $ in the ratio $ 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. [10] [11] 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. [10] [11] In a vector space , the midpoint between vectors $ \vec{u} $ and $ \vec{v} $ is $ \frac{\vec{u} + \vec{v}}{2} $, which applies to displacement calculations; for example, if $ \vec{u} $ and $ \vec{v} $ are position vectors from a reference origin, this average represents the displacement to the midpoint, useful in analyzing symmetric motion or balancing forces in mechanics . [10] This vector approach generalizes the coordinate-based formula to arbitrary dimensions without relying on a specific basis. [11] Construction Techniques Compass and Straightedge Method The compass and straightedge method is a classical technique in Euclidean geometry for constructing the midpoint of a given line segment , relying solely on these two tools to ensure precision without measurement. The compass allows for drawing circles of specified radii, while the straightedge 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 ancient Greek mathematics . Historically, the method originates from Euclid '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 perpendicular 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 compass and straightedge , establishing a rigorous procedure that has influenced geometric practice for over two millennia. [7] To construct the midpoint of a line segment AB using this method, begin by drawing the segment AB with the straightedge . Place the compass point at A and adjust the radius to the length of AB, then draw a circle (or arc) centered at A that passes through B; repeat the process with the compass centered at B, drawing another circle of the same radius that passes through A. These two circles will intersect at two points, say C and D, which are equidistant from A and B. Use the straightedge to draw the line connecting C and D, forming the perpendicular bisector of AB. The point where this bisector intersects AB is the midpoint M. This construction works because any point on the perpendicular bisector is equidistant from A and B by the properties of circles. [7] [12] The method assumes the line segment is straight and finite, as defined in Euclidean geometry , 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 compass and straightedge 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 computation within a Cartesian coordinate system , 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 ) (x_1, y_1) ( x 1 ​ , y 1 ​ ) and ( x 2 , y 2 ) (x_2, y_2) ( x 2 ​ , y 2 ​ ) . The midpoint M M M is then found by averaging these coordinates using the formula: M = ( x 1 + x 2 2 , y 1 + y 2 2 ) . M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). M = ( 2 x 1 ​ + x 2 ​ ​ , 2 y 1 ​ + y 2 ​ ​ ) . This averaging process yields the precise location of the midpoint, which can be plotted on a graph or incorporated into subsequent geometric calculations. [2] [13] 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