Varignon's theorem
Varignon's theorem is a fundamental result in Euclidean geometry stating that connecting the midpoints of the sides of any quadrilateral forms a parallelogram, known as the Varignon parallelogram.[1] This holds true for convex, concave, and even self-intersecting quadrilaterals, though the resulting parallelogram may appear degenerate in certain cases.[2] The theorem is named after the French mathematician and priest Pierre Varignon (1654–1722), whose proof was published posthumously in 1731 as part of his work Elémens de mathématique.[3] Despite its late publication relative to classical geometry, the result is remarkably simple and elegant, as noted by H.S.M. Coxeter and S.L. Greitzer in their 1967 text Geometry Revisited, where they express surprise at its relatively recent discovery.[1] Key properties of the Varignon parallelogram include its area being exactly half that of the original quadrilateral and its perimeter equaling the sum of the lengths of the quadrilateral's diagonals.[1] Each pair of opposite sides in the parallelogram is parallel to one of the quadrilateral's diagonals and equal in length to half that diagonal, which directly accounts for the perimeter relationship.[3] These attributes make the theorem useful in vector geometry, coordinate proofs, and applications such as computer graphics for simplifying quadrilateral transformations.[4]Statement
Formal Statement
A quadrilateral is a four-sided polygon defined by four vertices A, B, C, and D connected in cyclic order by line segments AB, BC, CD, and DA. The midpoint of a line segment joining two points with position vectors \vec{A} and \vec{B} is the point M with position vector \vec{M} = \frac{\vec{A} + \vec{B}}{2}. Let M, N, P, and Q denote the midpoints of sides AB, BC, CD, and DA, respectively, so that \vec{N} = \frac{\vec{B} + \vec{C}}{2}, \quad \vec{P} = \frac{\vec{C} + \vec{D}}{2}, \quad \vec{Q} = \frac{\vec{D} + \vec{A}}{2}. Varignon's theorem states that the quadrilateral MNPQ formed by joining these midpoints in order is a parallelogram.[1]Geometric Interpretation
Varignon's theorem provides an intuitive geometric transformation: no matter the shape of a quadrilateral—whether convex, concave, or crossed—connecting the midpoints of its sides always produces a parallelogram. This outcome underscores the theorem's universality, turning an arbitrary four-sided figure into a more symmetric one with inherent parallel properties.[2] At its core, the midpoints serve as averaging points between adjacent vertices, which geometrically translates to the new sides being parallel to the diagonals of the original quadrilateral and half their length. This averaging effect ensures that opposite sides of the resulting figure are equal and parallel, embodying the essence of a parallelogram without regard to the quadrilateral's initial irregularities.[3] To visualize this, consider an irregular trapezoid with one pair of parallel sides of unequal length and non-parallel legs differing in slope; linking the midpoints of its sides forms a parallelogram where the opposite sides match in both direction and magnitude, illustrating how the theorem simplifies complex shapes into balanced forms.[2] A key aspect of this interpretation is the shared centrality: the center of the Varignon parallelogram aligns precisely with the centroid of the original quadrilateral, representing the average position of its vertices.[5]Historical Background
Discovery and Attribution
Varignon's theorem is attributed to the French mathematician and priest Pierre Varignon (1654–1722), who provided the first explicit and rigorous proof of the result.[6][7] The theorem appeared posthumously in 1731, nine years after Varignon's death, within the textbook Élémens de mathématiques, a comprehensive secondary-level work on mathematics compiled from his lecture notes by colleagues at the Collège Royal in Paris.[6][7] This publication, spanning geometry among other topics, presented the theorem as a corollary to properties of lines intersecting triangle sides, emphasizing its pedagogical value in Euclidean geometry.[7] Varignon's discovery emerged amid his broader scholarly pursuits in mechanics and statics, where parallelogram constructions were instrumental for analyzing force compositions and equilibria.[6] His earlier works, such as Projet d'une nouvelle mécanique (1687) and Nouvelle mécanique ou statique (1725), laid groundwork in geometrical statics that aligned with the theorem's implications for vector-like representations in physical systems.[6]Development and Recognition
During the 19th century, Varignon's theorem gained prominence through its inclusion in geometry textbooks that supported the revival of Euclidean methods, reflecting the period's emphasis on synthetic proofs and quadrilateral properties. This recognition aligned with broader efforts to systematize classical geometry, as seen in foundational texts like Adrien-Marie Legendre's Éléments de géométrie (1794). The theorem's utility in demonstrating parallelogram formation from arbitrary quadrilaterals made it a staple in educational materials, contributing to its widespread adoption in mathematical instruction. The theorem's development accelerated in the 19th century with the emergence of vector geometry, popularized by William Rowan Hamilton's quaternions (1843) and J. Willard Gibbs's vector analysis (1881–1884). These methods offered a natural vector-based proof of the theorem, where the midpoints of the quadrilateral's sides are expressed as average position vectors, and their connections form a parallelogram via the parallelogram law of vector addition. This approach simplified demonstrations and integrated the theorem into the growing field of analytical geometry, influencing subsequent texts on vector methods.[8] In the 20th century, the theorem's extensions linked it to affine transformations, where the Varignon parallelogram remains a parallelogram under affine mappings, underscoring its role in preserving midpoint structures and parallel lines. This connection has been highlighted in modern geometric analyses, such as projective analogues that generalize the theorem to higher dimensions and non-Euclidean settings.[9] Additionally, in computational geometry, the theorem aids in algorithms for polygon simplification and mesh generation, where connecting midpoints facilitates efficient computation of parallelogram approximations for arbitrary quadrilaterals. Notable mentions of the theorem appear in influential works on quadrilateral properties, and H.S.M. Coxeter's Geometry Revisited (1967), where it is employed to explore cyclic and complete quadrangles. These contributions underscore the theorem's enduring impact in both classical and advanced geometric literature.[10]Proofs
Vector-Based Proof
Consider a quadrilateral ABCD with vertices assigned position vectors \vec{A}, \vec{B}, \vec{C}, and \vec{D} relative to an arbitrary origin. Let M, N, P, and Q denote the midpoints of sides AB, BC, CD, and DA, respectively. The position vector of M is \vec{M} = \frac{\vec{A} + \vec{B}}{2}, of N is \vec{N} = \frac{\vec{B} + \vec{C}}{2}, of P is \vec{P} = \frac{\vec{C} + \vec{D}}{2}, and of Q is \vec{Q} = \frac{\vec{D} + \vec{A}}{2}. To show that MNPQ is a parallelogram, examine the vectors representing opposite sides. The vector \overrightarrow{MN} = \vec{N} - \vec{M} = \frac{\vec{B} + \vec{C}}{2} - \frac{\vec{A} + \vec{B}}{2} = \frac{\vec{C} - \vec{A}}{2}. Similarly, \overrightarrow{QP} = \vec{P} - \vec{Q} = \frac{\vec{C} + \vec{D}}{2} - \frac{\vec{D} + \vec{A}}{2} = \frac{\vec{C} - \vec{A}}{2}. Thus, \overrightarrow{MN} = \overrightarrow{QP}, implying that sides MN and QP are equal in magnitude and parallel. Analogously, the vector \overrightarrow{NP} = \vec{P} - \vec{N} = \frac{\vec{C} + \vec{D}}{2} - \frac{\vec{B} + \vec{C}}{2} = \frac{\vec{D} - \vec{B}}{2}, and \overrightarrow{MQ} = \vec{Q} - \vec{M} = \frac{\vec{D} + \vec{A}}{2} - \frac{\vec{A} + \vec{B}}{2} = \frac{\vec{D} - \vec{B}}{2}. Therefore, \overrightarrow{NP} = \overrightarrow{MQ}, confirming that the remaining pair of opposite sides is also equal and parallel. This establishes that MNPQ is a parallelogram.[11] This vector approach is coordinate-free, relying solely on vector addition and scalar multiplication, and applies in any affine space, including higher dimensions for skew quadrilaterals.Coordinate Geometry Proof
To prove Varignon's theorem using coordinate geometry, consider an arbitrary quadrilateral ABCD in the Euclidean plane with vertices assigned coordinates A(x_1, y_1), B(x_2, y_2), C(x_3, y_3), and D(x_4, y_4), where the points are in general position and the quadrilateral may be convex or non-convex.[12] Denote the midpoints of the sides as follows:M, the midpoint of AB, has coordinates M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right); N, the midpoint of BC, has coordinates N = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right); P, the midpoint of CD, has coordinates P = \left( \frac{x_3 + x_4}{2}, \frac{y_3 + y_4}{2} \right); Q, the midpoint of DA, has coordinates Q = \left( \frac{x_4 + x_1}{2}, \frac{y_4 + y_1}{2} \right). The quadrilateral MNPQ formed by connecting these midpoints in order must be shown to be a parallelogram, meaning its opposite sides are parallel and equal in length.[12] To verify parallelism, compute the slopes of the sides. The slope of MN is m_{MN} = \frac{\frac{y_2 + y_3}{2} - \frac{y_1 + y_2}{2}}{\frac{x_2 + x_3}{2} - \frac{x_1 + x_2}{2}} = \frac{(y_3 - y_1)/2}{(x_3 - x_1)/2} = \frac{y_3 - y_1}{x_3 - x_1}, provided x_3 \neq x_1; if x_3 = x_1, both MN and the relevant diagonal are vertical and thus parallel. The slope of QP is m_{QP} = \frac{\frac{y_3 + y_4}{2} - \frac{y_4 + y_1}{2}}{\frac{x_3 + x_4}{2} - \frac{x_4 + x_1}{2}} = \frac{(y_3 - y_1)/2}{(x_3 - x_1)/2} = \frac{y_3 - y_1}{x_3 - x_1}. Since m_{MN} = m_{QP}, sides MN and QP are parallel.[12] Similarly, the slope of NP is m_{NP} = \frac{\frac{y_3 + y_4}{2} - \frac{y_2 + y_3}{2}}{\frac{x_3 + x_4}{2} - \frac{x_2 + x_3}{2}} = \frac{(y_4 - y_2)/2}{(x_4 - x_2)/2} = \frac{y_4 - y_2}{x_4 - x_2}, provided x_4 \neq x_2; vertical cases follow analogously. The slope of QM is m_{QM} = \frac{\frac{y_1 + y_2}{2} - \frac{y_4 + y_1}{2}}{\frac{x_1 + x_2}{2} - \frac{x_4 + x_1}{2}} = \frac{(y_2 - y_4)/2}{(x_2 - x_4)/2} = \frac{y_2 - y_4}{x_2 - x_4} = \frac{y_4 - y_2}{x_4 - x_2}. Thus, m_{NP} = m_{QM}, so NP and QM are parallel. With both pairs of opposite sides parallel, MNPQ is a parallelogram.[12] To confirm equal lengths, apply the distance formula. The length of MN is |MN| = \sqrt{ \left( \frac{x_3 - x_1}{2} \right)^2 + \left( \frac{y_3 - y_1}{2} \right)^2 } = \frac{1}{2} \sqrt{ (x_3 - x_1)^2 + (y_3 - y_1)^2 }, which is half the length of diagonal AC. The differences in coordinates for QP yield the same expression: |QP| = \sqrt{ \left( \frac{x_3 - x_1}{2} \right)^2 + \left( \frac{y_3 - y_1}{2} \right)^2 } = \frac{1}{2} \sqrt{ (x_3 - x_1)^2 + (y_3 - y_1)^2 }. Thus, |MN| = |QP|. For the other pair, |NP| = \frac{1}{2} \sqrt{ (x_4 - x_2)^2 + (y_4 - y_2)^2 }, half the length of diagonal BD, and similarly |QM| = |NP|. The equal lengths of opposite sides reinforce that MNPQ is a parallelogram. This approach holds for quadrilaterals in general position, including non-convex cases where the midpoints remain well-defined and the resulting figure satisfies the parallelogram properties.[12]