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Quadrilateral

A quadrilateral is a polygon in the Euclidean plane consisting of four edges (or sides) connected by four vertices (or corners). Quadrilaterals are fundamental two-dimensional shapes in geometry, distinguished from triangles by having one more side and from pentagons by having one fewer. The sum of the interior angles of any simple quadrilateral is always 360 degrees.^[1] Quadrilaterals can be classified as simple or complex, with simple quadrilaterals further divided into convex and concave subtypes based on their interior angles and the position of their diagonals. A convex quadrilateral has all interior angles less than 180 degrees, with both diagonals lying entirely within the shape. In contrast, a concave quadrilateral (also called a dart or arrowhead) features one interior angle greater than 180 degrees, causing one diagonal to lie outside the figure. Complex quadrilaterals, such as crossed trapezoids or antiparallelograms, are self-intersecting and do not form a simple closed shape without overlapping sides.^[1] Among convex quadrilaterals, several special types are defined by specific side and angle properties, forming a hierarchical structure. A trapezoid in North American English (or trapezium in British English, where the terms are sometimes reversed) has at least one pair of parallel sides. A parallelogram has two pairs of parallel sides, opposite sides equal in length, and opposite angles equal. Further specializations include the rectangle (a parallelogram with four right angles), the rhombus (a parallelogram with all sides equal), and the square (a rectangle or rhombus with both properties). The kite, another distinct type, has two pairs of adjacent equal sides and often features perpendicular diagonals, one of which bisects the other.^[1] Key properties of quadrilaterals include their diagonals, which connect non-adjacent vertices and can bisect each other or form right angles in certain subtypes. The area of a quadrilateral varies by type; for example, a parallelogram's area is base times height or ab \sin \theta where a and b are adjacent sides and \theta is the included angle, while a general convex quadrilateral uses \frac{1}{2}pq \sin \theta with diagonals p and q and the angle \theta between them. These shapes appear in architecture and design, such as in building models and structural nets.^[1]^[2]

Definition and Classifications

Definition

A quadrilateral is a polygon consisting of four edges (or sides) and four vertices, forming a closed two-dimensional figure in a plane unless otherwise specified.^[3] As a specific case of a polygon—a plane figure bounded by a finite chain of line segments closing in a loop— a quadrilateral has exactly four such segments connecting its vertices in sequence.^[4]^[5] Quadrilaterals are classified as simple or complex based on whether their sides intersect. A simple quadrilateral has no self-intersecting sides, maintaining a single boundary without crossings, whereas a complex quadrilateral features sides that intersect themselves, creating additional regions within the figure.^[6]^[7] The term "quadrilateral" originates from the Latin words quadri- meaning "four" and lateralis meaning "sided," reflecting its four-sided nature.^[8] Early discussions of such figures appear in Euclidean geometry, with Euclid's Elements around 300 BCE describing types like squares and rhombi as quadrilateral figures in Book I.^[9] A representative example is the square, where all four sides and angles are equal; in contrast, an irregular quadrilateral might have sides of varying lengths and unequal angles, such as a shape with sides measuring 3, 4, 5, and 6 units.^[10]

Convex Quadrilaterals

A convex quadrilateral is a four-sided polygon in which all interior angles measure less than 180 degrees, ensuring the figure does not "cave in" or reenter itself. This property distinguishes it as the standard form for many geometric theorems, where the polygon remains on one side of each of its sides when extended. Both diagonals connecting non-adjacent vertices lie entirely within the interior of the quadrilateral, facilitating straightforward applications in proofs and constructions.^[11]^[12] The sum of the interior angles of a convex quadrilateral is always 360 degrees. This result follows from dividing the quadrilateral into two triangles by drawing one diagonal; each triangle has an interior angle sum of 180 degrees, yielding a total of 360 degrees for the quadrilateral. Additionally, the sum of the exterior angles—one at each vertex, measured in the direction of traversal—equals 360 degrees, a property shared with all convex polygons regardless of the number of sides.^[13]^[14] For the side lengths a, b, c, and d of a convex quadrilateral to form a closed figure, the sum of any three sides must exceed the length of the fourth side. This quadrilateral inequality implies that each individual side is strictly less than the semiperimeter s = (a + b + c + d)/2. Unlike concave quadrilaterals, which feature one reflex interior angle exceeding 180 degrees, convex ones maintain this non-reentrant structure essential for properties like diagonal containment.^[15] A representative example is the rectangle, a convex quadrilateral with all four interior angles measuring exactly 90 degrees, thereby summing to 360 degrees. In a rectangle, the diagonals not only lie inside but also bisect each other and are equal in length, underscoring the symmetry inherent in many convex forms.^[12]

Concave Quadrilaterals

A concave quadrilateral is a simple quadrilateral in which exactly one interior angle measures greater than 180°, creating a reflex angle that causes the figure to dent inward at that vertex, giving it a reentrant shape.^[11]^[16] This inward indentation distinguishes concave quadrilaterals from convex ones, where all interior angles are less than 180° and the figure lies entirely on one side of each of its sides.^[11] The sum of the interior angles in a concave quadrilateral remains 360°, consistent with all simple quadrilaterals, but the presence of the reflex angle means that one diagonal connects vertices such that part of it lies outside the boundary of the polygon.^[17] This external diagonal segment arises because the reflex vertex "pushes" one diagonal beyond the enclosed region, impacting properties like the quadrilateral's ability to fully enclose space without protrusion.^[10] A representative example is the dart, also known as an arrowhead, which is a concave quadrilateral with bilateral symmetry similar to a kite but featuring a reflex angle at one vertex. In the dart's vertex configuration, two pairs of adjacent sides are equal in length, with the reflex angle typically at the indented "tail" vertex, while the other three angles are acute or obtuse, summing to less than 180° collectively to maintain the total of 360°.^[11]^[18]

Self-Intersecting Quadrilaterals

A self-intersecting quadrilateral is a four-sided polygon in which two non-adjacent sides cross each other at a point that is not a vertex, resulting in a figure that violates the non-intersection property of simple polygons.^[3] Also referred to as a crossed quadrilateral, bow-tie quadrilateral, or butterfly quadrilateral, this configuration forms shapes resembling a bow-tie or a simple star-like structure.^[1] Unlike convex or concave quadrilaterals, which maintain a simple boundary, the self-intersection occurs when the quadrilateral is traversed in sequential vertex order, causing the edges to overlap in the plane.^[3] This crossing divides the plane into three distinct regions rather than the two (interior and exterior) produced by a simple closed curve, as the intersection point creates an additional bounded area amid the overlapping sections.^[19] Self-intersecting quadrilaterals are not classified as simple polygons because their boundaries intersect themselves, complicating topological properties and preventing them from tiling the plane without gaps or overlaps in the manner of non-crossed quadrilaterals.^[3] For instance, in a typical crossed quadrilateral ABCD, sides AD and BC intersect at an interior point, forming two triangular lobes connected at the crossing.^[1] Area determination for these figures requires decomposition into non-overlapping triangles, often employing oriented (algebraic) area calculations where regions traversed in opposite directions contribute positively or negatively to avoid double-counting the intersection zone.^[19] This approach contrasts with the direct formulas applicable to simple quadrilaterals, highlighting the need for careful subdivision to capture the true enclosed space. In relation to other types, self-intersecting quadrilaterals differ from concave ones by featuring actual edge crossings rather than mere indentations.^[3]

Skew Quadrilaterals

A skew quadrilateral is a four-sided polygonal chain in three-dimensional Euclidean space whose vertices are not all coplanar.^[20] Unlike planar quadrilaterals, which lie entirely within a single plane, the points of a skew quadrilateral span a volume in 3D space, with its sides potentially intersecting or remaining separate depending on their configuration.^[20] This non-planar nature makes the skew quadrilateral analogous to skew lines in 3D geometry, where two lines neither intersect nor are parallel and do not lie in the same plane.^[21] No single plane contains all four vertices, emphasizing its spatial extension beyond two dimensions. Planar quadrilaterals represent the special coplanar case of this more general figure.^[3] A representative example of a skew quadrilateral is formed by selecting two skew lines as opposite sides and connecting their respective endpoints with two transversal line segments, creating a closed but non-planar chain.^[22] Such constructions arise in space geometry, for instance, where edges of polyhedra like tetrahedra or octahedra form skew quadrilaterals that influence properties such as midlines or dihedral angles.^[1] These figures are studied in solid geometry to explore 3D configurations and their implications for volumes and intersections in polyhedral structures.^[23]

Basic Geometric Properties

Sides and Angles

A quadrilateral consists of four sides, typically denoted by lengths a, b, c, and d. The perimeter P of a quadrilateral is the total length of its boundary, calculated as the sum of these side lengths: P = a + b + c + d.^[24] For any simple quadrilateral—meaning one that does not intersect itself—the sum of the interior angles is $360^\circ. This result follows from the angle sum theorem for quadrilaterals, proved by drawing one diagonal to divide the figure into two triangles, each with an interior angle sum of $180^\circ, yielding a total of $360^\circ.^[25]^[26] A notable property relating to the sides is Varignon's theorem, which states that the quadrilateral formed by connecting the midpoints of the four sides of any quadrilateral is a parallelogram.^[27] Certain quadrilaterals exhibit special relationships between their sides and angles. A cyclic quadrilateral has all four vertices lying on a single circle, while a tangential quadrilateral has all four sides tangent to a single inscribed circle. These configurations introduce additional constraints on angle measures and side lengths, explored further in classifications of quadrilaterals.^[28]^[29] The diagonals of a quadrilateral connect opposite vertices and play a role in dividing the interior angles indirectly through triangulation.^[30]

Diagonals

A diagonal of a quadrilateral is a line segment joining two non-adjacent vertices.^[3] In a simple quadrilateral, the two diagonals divide the figure into two triangles.^[31] The position and intersection of diagonals vary by quadrilateral type. In a convex quadrilateral, both diagonals lie entirely within the figure and intersect at a single interior point.^[31] In a concave quadrilateral, one diagonal lies partially outside the figure, and the diagonals may intersect outside the boundary.^[31] In a self-intersecting quadrilateral, the diagonals may coincide with sides or extend outside the crossed configuration.^[31] For cyclic quadrilaterals, Ptolemy's theorem relates the diagonals to the sides: if the sides are a, b, c, d and the diagonals are p, q, then pq = ac + bd.^[32] A fundamental identity holds for any convex quadrilateral with sides a, b, c, d and diagonals p, q: the sum of the squares of the sides equals the sum of the squares of the diagonals plus four times the square of the segment joining the midpoints of the diagonals, or

a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4m^2,

where m is that midpoint segment (analogous to bimedians formed by midpoints of sides).^[3]

Special Line Segments

In tangential quadrilaterals, the four interior angle bisectors are concurrent at a single point known as the incenter, which serves as the center of the incircle tangent to all four sides.^[33] This concurrence provides a geometric characterization: a convex quadrilateral admits an incircle if and only if its angle bisectors meet at one point.^[34] The lengths of these bisectors can be determined by adapting the angle bisector theorem from triangles, considering the division of opposite sides in ratios related to the adjacent side lengths and the tangential condition where the sums of opposite sides are equal.^[35] For cyclic quadrilaterals, which possess a circumcircle passing through all four vertices, the perpendicular bisectors of the four sides intersect concurrently at the circumcenter, the center of that circumcircle. This property holds if and only if the quadrilateral is cyclic, enabling the construction of the circumcircle via these bisectors, analogous to the process in triangles. In general quadrilaterals, the Newton line is a constructible segment joining the midpoints of the two diagonals, with additional significance in special cases; for tangential quadrilaterals, the incenter lies on this line, linking the concurrency of angle bisectors to the diagonals' midpoints.^[36]^[37] A representative example occurs in the rhombus, where the diagonals themselves act as the angle bisectors, aligning with and bisecting all four vertex angles due to the equal side lengths and perpendicular diagonals.^[38] This alignment simplifies constructions, as the diagonals divide the rhombus into congruent triangles where each half-angle is bisected by the intersection point.^[39]

Area Formulas

Trigonometric Formulas

For a convex quadrilateral, one common trigonometric approach to finding the area involves dividing the figure into two triangles by drawing one diagonal, then summing their areas using the standard triangle area formula \frac{1}{2}xy \sin Z, where x and y are two sides enclosing angle Z. Suppose the quadrilateral has sides a, b, c, d in sequence and one diagonal divides it into two triangles with included angles B and D at the vertices opposite the diagonal. The area K is then K = \frac{1}{2} ab \sin B + \frac{1}{2} cd \sin D.^[40] This derivation relies on the fact that the diagonal splits the quadrilateral into two non-overlapping triangles whose areas add directly, assuming the quadrilateral is convex so the diagonal lies inside.^[40] An alternative trigonometric formula uses both diagonals p and q, which intersect at angle \theta in a convex quadrilateral. The area is given by

K = \frac{1}{2} p q \sin \theta.

This follows from splitting the quadrilateral into four triangles formed by the intersecting diagonals, where opposite pairs have equal areas \frac{1}{2} (p_1 q_1 \sin \theta) + \frac{1}{2} (p_2 q_2 \sin \theta) = \frac{1}{2} p q \sin \theta, with p = p_1 + p_2 and q = q_1 + q_2.^[3] The formula assumes the intersection point divides the diagonals into segments whose products yield the total, and \sin \theta > 0 for acute or obtuse \theta within the convex shape.^[3] For a cyclic quadrilateral, where opposite angles sum to \pi, the area simplifies via Brahmagupta's formula:

K = \sqrt{(s - a)(s - b)(s - c)(s - d)},

with semiperimeter s = \frac{1}{2}(a + b + c + d). This non-trigonometric expression derives trigonometrically from Bretschneider's general formula by substituting A + B = \pi, making \cos^2 \left( \frac{A + B}{2} \right) = 0.^[41] The general trigonometric formula for any quadrilateral (convex or concave) is Bretschneider's formula:

K = \sqrt{(s - a)(s - b)(s - c)(s - d) - abcd \cos^2 \left( \frac{A + B}{2} \right)},

where A and B are one pair of opposite angles. This extends the cyclic case and requires knowledge of at least two opposite angles or equivalent trigonometric data; without angles, non-trigonometric methods are needed.^[42] These formulas prioritize cases where angles or diagonal intersections are measurable, providing exact areas when sides alone are insufficient.^[42]

Non-Trigonometric Formulas

Non-trigonometric formulas for the area of a quadrilateral rely on side lengths, diagonals, or vertex coordinates, avoiding direct computation of angles. These methods are particularly useful for cyclic quadrilaterals, where the area depends solely on the sides, and for general cases using additional geometric elements like diagonals. For coordinate representations, the shoelace formula provides a straightforward calculation applicable to both convex and concave quadrilaterals. For a cyclic quadrilateral with side lengths a, b, c, and d, Brahmagupta's formula gives the area directly from the sides. Named after the Indian mathematician Brahmagupta (7th century CE), it states that the area K is

K = \sqrt{(s - a)(s - b)(s - c)(s - d)},

where s = \frac{a + b + c + d}{2} is the semiperimeter. This formula generalizes Heron's formula for triangles and applies only to quadrilaterals that can be inscribed in a circle.^[41] Bretschneider's formula extends this to general convex quadrilaterals, incorporating the sum of opposite angles \alpha and \gamma. The area K is

K = \sqrt{(s - a)(s - b)(s - c)(s - d) - abcd \cos^2 \left( \frac{\alpha + \gamma}{2} \right)},

where s is again the semiperimeter. Developed by Carl Anton Bretschneider in 1842, this reduces to Brahmagupta's formula when the quadrilateral is cyclic, as opposite angles sum to $180^\circ and \cos(90^\circ) = 0. An equivalent non-trigonometric form expresses the area in terms of the sides a, b, c, d and diagonals p, q:

K = \frac{1}{4} \sqrt{4p^2 q^2 - (b^2 + d^2 - a^2 - c^2)^2}.

This version, derived via vector methods, avoids angles entirely and applies to convex quadrilaterals.^[42] The shoelace formula, also known as Gauss's area formula, computes the area using the Cartesian coordinates of the vertices. For a quadrilateral with vertices (x_1, y_1), (x_2, y_2), (x_3, y_3), and (x_4, y_4) listed in clockwise or counterclockwise order, the area A is

A = \frac{1}{2} \left| x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1 - (y_1 x_2 + y_2 x_3 + y_3 x_4 + y_4 x_1) \right|.

This formula originates from 18th-19th century surveying techniques and can be derived by decomposing the quadrilateral into two triangles sharing a diagonal, then applying the determinant form for each triangle's area (which is \frac{1}{2} | (x_i y_{i+1} - x_{i+1} y_i) |), and summing the results. Alternatively, it follows from Green's theorem in the plane, integrating over the boundary. The absolute value ensures a positive area regardless of ordering direction.^[43] For concave quadrilaterals, these non-trigonometric formulas remain applicable with adjustments. The shoelace formula yields the correct area when vertices are ordered sequentially around the boundary, accounting for the indentation via signed areas that cancel appropriately. For side- or diagonal-based formulas like Bretschneider's or Brahmagupta's (if adaptable), one can divide the shape into a triangle and a pentagon or subtract the area of the external triangular region protruding beyond the concave vertex, computed separately using the shoelace method or Heron's formula on the relevant triangles. Bretschneider's formula directly supports concave cases without modification.^[42]

Vector-Based Formulas

Vector-based formulas provide a unified framework for computing properties of quadrilaterals, particularly the area, using the cross product of diagonal vectors. For a quadrilateral ABCD with diagonals represented as vectors \overrightarrow{AC} and \overrightarrow{BD}, the vector area is given by \frac{1}{2} \overrightarrow{AC} \times \overrightarrow{BD}. The magnitude of this vector yields the scalar area A = \frac{1}{2} \|\overrightarrow{AC} \times \overrightarrow{BD}\|, which holds for any planar quadrilateral regardless of convexity.^[44] This formula arises from decomposing the quadrilateral into two triangles sharing a diagonal intersection and summing their vector areas, resulting in the cross product of the diagonals scaled by half. In two dimensions, it specializes to the shoelace formula when coordinates are used. For skew quadrilaterals in three-dimensional space, where the vertices are non-coplanar, the Varignon parallelogram—formed by connecting the midpoints of the sides—remains planar, and its area is \frac{1}{4} \|\overrightarrow{AC} \times \overrightarrow{BD}\|. Consequently, the effective area of the skew quadrilateral is defined as twice the area of this Varignon parallelogram, yielding the same magnitude \frac{1}{2} \|\overrightarrow{AC} \times \overrightarrow{BD}\| via vector sums of the side midpoints.^[45]

Advanced Metric Properties

Lengths of Diagonals and Bimedians

In a general quadrilateral ABCD with sides AB = a, BC = b, CD = c, DA = d, and diagonals AC = p, BD = q, the length of a diagonal can be determined by dividing the quadrilateral into two triangles and applying the law of cosines. For diagonal p = AC, in triangle ABC, p^2 = a^2 + b^2 - 2ab \cos B, and in triangle ADC, p^2 = d^2 + c^2 - 2dc \cos D. Since the interior angles B and D are not necessarily related without additional conditions, the expression requires knowledge of one angle or other parameters.^[46] The bimedians of a quadrilateral are the line segments connecting the midpoints of opposite sides. For the bimedian m_a joining the midpoints of sides a and c, its length is given by m_a = \frac{1}{2} \sqrt{2b^2 + 2d^2 - a^2 - c^2}. Similarly, the bimedian m_b joining the midpoints of sides b and d has length m_b = \frac{1}{2} \sqrt{2a^2 + 2c^2 - b^2 - d^2}. These formulas arise from coordinate or vector geometry applied to the midpoints and are particularly useful in constructing the Varignon parallelogram, where the bimedians serve as its diagonals.^[47] For cyclic quadrilaterals, Ptolemy's theorem provides a direct relation between the sides and diagonals: the product of the diagonals equals the sum of the products of the opposite sides, or pq = ac + bd. This theorem, originally stated by Ptolemy in the 2nd century AD, holds if and only if the quadrilateral is cyclic and can be proved using similar triangles or the law of cosines on the inscribed figure.^[32] Euler's quadrilateral theorem further elaborates this relation: the sum of the squares of the sides equals the sum of the squares of the diagonals plus four times the square of the segment joining the midpoints of the diagonals, or a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4m^2, where m is the length of that midpoint segment. Proved by Euler in 1748 using geometric decompositions and the cosine law, this identity quantifies the "twist" in non-parallelogram quadrilaterals and equals the parallelogram law when m = 0.^[48]

Trigonometric Identities

In a quadrilateral ABCD with sides AB = d, BC = a, CD = b, DA = c, and diagonals p = AC, q = BD, the law of cosines can be applied to the triangles formed by each diagonal to relate sides, angles, and diagonals. Specifically, splitting the quadrilateral along diagonal p yields triangles ABC and ADC. In \triangle ABC, the law of cosines gives

p^2 = a^2 + d^2 - 2ad \cos B,

while in \triangle ADC,

p^2 = b^2 + c^2 - 2bc \cos D.

Equating these expressions produces the trigonometric identity

a^2 + d^2 - 2ad \cos B = b^2 + c^2 - 2bc \cos D,

which connects the sides and opposite angles B and D. A similar relation holds for the other diagonal q, splitting into \triangle ABD and \triangle BCD:

q^2 = d^2 + c^2 - 2dc \cos A = a^2 + b^2 - 2ab \cos C.

These identities hold for any convex quadrilateral and provide a foundation for deriving further metric properties, such as diagonal lengths in terms of sides and angles.^[49] For cyclic quadrilaterals, where all vertices lie on a common circle, the opposite angles sum to 180^\circ, so A + C = 180^\circ and B + D = 180^\circ. This supplementary property implies immediate trigonometric equalities, such as \sin A = \sin C and \sin B = \sin D, since \sin(180^\circ - \theta) = \sin \theta. Similarly, \cos A = -\cos C and \cos B = -\cos D. Substituting into the general law of cosines identity for diagonal p yields

a^2 + d^2 - 2ad \cos B = b^2 + c^2 + 2bc \cos B,

which rearranges to

(a^2 + d^2 - b^2 - c^2) = 2\cos B (ad + bc).

This specialized form is useful for computing diagonals or verifying cyclicity. Additionally, the half-angle relation \tan(A/2) \tan(C/2) = 1 holds, as A/2 + C/2 = 90^\circ implies \tan(C/2) = \cot(A/2), and a symmetric equality applies to B and D. These identities distinguish cyclic quadrilaterals and underpin theorems like Ptolemy's.^[50]^[51] Such trigonometric identities extend to applications like area computation via Bretschneider's formula, which generalizes the cyclic case but incorporates \cos((B+D)/2). For tangential quadrilaterals with an incircle of radius r and semiperimeter s, the area is simply K = r s, linking back to trigonometric expressions for r in terms of angles when sides are known.^[52]

Inequalities

In convex quadrilaterals, the side lengths a, b, c, d must satisfy the generalized triangle inequality: the sum of any three sides is strictly greater than the fourth side, ensuring the figure is non-degenerate and can be formed without collapsing into a line segment. This condition arises from applying the triangle inequality to the two triangles formed by one diagonal in the quadrilateral. For example, a + b + c > d, a + b + d > c, a + c + d > b, and b + c + d > a. Equality holds only in degenerate cases where the quadrilateral flattens. The diagonals p and q also obey specific inequalities relative to the sides. The sum of the diagonals is less than the perimeter: p + q < a + b + c + d. This follows from the triangle inequality applied to the four triangles created by the intersecting diagonals. Additionally, the sum of each pair of opposite sides is less than the sum of the diagonals: a + c < p + q and b + d < p + q, with equality in limiting cases like when the quadrilateral degenerates into a line. These bounds highlight the diagonals' role in spanning the figure more efficiently than opposite sides.^[48] For the bimedians—the line segments joining the midpoints of opposite sides, denoted m and n—inequalities can be derived from their length formulas. The length of the bimedian connecting the midpoints of sides a and c is given by

m = \frac{1}{2} \sqrt{2b^2 + 2d^2 - a^2 - c^2},

and similarly for the other bimedian n = \frac{1}{2} \sqrt{2a^2 + 2c^2 - b^2 - d^2}. For positivity, these imply a^2 + c^2 < 2(b^2 + d^2) and b^2 + d^2 < 2(a^2 + c^2), providing bounds on pairs of opposite sides in terms of the adjacent pairs. A related inequality is that the sum of the bimedians satisfies m + n \geq \max(a, c), reflecting their average positioning across the figure, though equality occurs in specific symmetric cases like rhombi. These relations connect to the Varignon parallelogram, where the bimedians are the diagonals scaled by half. Area inequalities provide bounds on the enclosed region K. The isoperimetric inequality for convex quadrilaterals states that p^2 \geq 16K, where p = a + b + c + d is the perimeter, with equality holding uniquely for the square. This adaptation of the classical isoperimetric problem shows that among all quadrilaterals with fixed perimeter, the square maximizes the area. For fixed side lengths, the maximum area is achieved by the cyclic quadrilateral, given by Brahmagupta's formula K = \sqrt{(s - a)(s - b)(s - c)(s - d)}, where s = p/2; any non-cyclic quadrilateral with the same sides has strictly smaller area.^[53]^[41] A key metric inequality analogous to Weitzenböck's for triangles is Euler's inequality for the sum of squares: a^2 + b^2 + c^2 + d^2 \geq p^2 + q^2, with equality if and only if the diagonals bisect each other (i.e., the quadrilateral is a parallelogram). This follows from the identity a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4t^2, where t is the distance between the midpoints of the diagonals, and t \geq 0. For the square, equality holds in the parallelogram case, maximizing efficiency in distributing lengths. This inequality bounds the diagonals in terms of the sides and underscores structural optima in convex figures.^[48]

Extremal and Structural Properties

Maximum and Minimum Properties

Among all quadrilaterals with given side lengths a, b, c, and d, the maximum area is achieved by the cyclic quadrilateral, where the vertices lie on a common circle. This result follows from the isoperimetric properties of cyclic polygons and can be computed using Brahmagupta's formula: \sqrt{(s - a)(s - b)(s - c)(s - d)}, where s = (a + b + c + d)/2 is the semiperimeter.^[28]^[54] Conversely, among all quadrilaterals with a fixed area A, the square has the minimum perimeter P. This is a consequence of the isoperimetric inequality for quadrilaterals, where the square maximizes the area for a given perimeter, implying the reverse extremum. For a square with area A, the side length is \sqrt{A} and perimeter $4\sqrt{A}.^[55]^[56] Regarding diagonals, in any convex quadrilateral, the sum of the lengths of the diagonals exceeds half the perimeter, with the infimum approached as the quadrilateral degenerates into a line segment. This bound arises from applying the triangle inequality to the triangles formed by each diagonal.^[57] For fixed side lengths, the sum of the squares of the diagonals p^2 + q^2 is minimized when the quadrilateral is a parallelogram. The general relation is p^2 + q^2 = a^2 + b^2 + c^2 + d^2 - 4t^2, where t is the distance between the midpoints of the diagonals; minimization occurs at t = 0, characteristic of parallelograms. In this case, p^2 + q^2 = 2(a^2 + c^2) assuming a = c and b = d as opposite sides.^[1] Finally, for a quadrilateral with fixed diagonal lengths p and q, the maximum area is \frac{1}{2} p q, achieved when the diagonals are perpendicular, forming an orthodiagonal quadrilateral. The area formula derives from dividing the quadrilateral into four right triangles at the intersection point, with the general area being \frac{1}{2} p q \sin \theta maximized at \theta = 90^\circ.^[58]

Remarkable Points and Lines

In a convex quadrilateral, the centroid is defined as the arithmetic mean of the coordinates of its four vertices, serving as the balance point if equal masses are placed at each vertex. This point coincides with the intersection of the two bimedians, where each bimedian—the line segment joining the midpoints of a pair of opposite sides—is bisected by the centroid in a 1:1 ratio.^[59]^[60] The Varignon parallelogram, constructed by connecting the midpoints of the sides of any quadrilateral in cyclic order, is always a parallelogram whose sides are parallel to the diagonals of the original quadrilateral and half their lengths. Its diagonals are the bimedians of the quadrilateral, and its area is exactly half that of the original figure. If the quadrilateral has perpendicular diagonals (an orthodiagonal quadrilateral), the Varignon parallelogram is a rectangle.^[27] The Newton line of a convex quadrilateral is the straight line passing through the midpoints of its two diagonals. In a parallelogram, these midpoints coincide, degenerating the line to a point; otherwise, the line measures the deviation from parallelogram properties. For a tangential quadrilateral (one admitting an incircle), the incenter lies on this Newton line, providing a geometric relation between the tangency points and the diagonal midpoints.^[37]^[61]

Taxonomy of Special Quadrilaterals

Special quadrilaterals are convex four-sided polygons distinguished by particular symmetries or relations among their sides, angles, or diagonals, forming a hierarchical taxonomy based on these defining properties. This classification builds upon the general convex quadrilateral by identifying subsets with enhanced geometric regularity, such as parallelism, equality of lengths or angles, and compatibility with circles. Common examples include parallelograms and their refinements, as well as figures defined by diagonal perpendicularity or tangential properties.^[3] An equilateral quadrilateral has all four sides of equal length and is specifically termed a rhombus. In a rhombus, opposite sides are parallel, and the diagonals bisect each other at right angles, providing a foundation for further specializations. An equiangular quadrilateral features all interior angles measuring 90 degrees and is known as a rectangle. Opposite sides in a rectangle are equal and parallel, ensuring right-angled corners throughout. A parallelogram is defined by having both pairs of opposite sides parallel and equal in length, with opposite angles equal and consecutive angles supplementary. Rectangles and rhombi are special parallelograms, inheriting these parallel properties while adding angular or side equalities.^[62] A trapezoid possesses exactly one pair of parallel sides, known as the bases, with the non-parallel sides called legs; an isosceles trapezoid further requires equal legs and equal base angles. Unlike parallelograms, only one pair of sides is parallel.^[63] A kite consists of two pairs of adjacent sides that are equal in length, resulting in diagonals that are perpendicular, with one diagonal serving as the axis of symmetry and bisecting the other. Kites often exhibit reflective symmetry along one diagonal. A cyclic quadrilateral is one that can be inscribed in a circle, such that all vertices lie on the circumference; a key property is that the sums of its opposite angles each equal 180 degrees. This inscription enables applications like Brahmagupta's formula for area computation.^[28] An orthodiagonal quadrilateral has diagonals that are perpendicular to each other, a condition met when the sums of the squares of opposite sides are equal (i.e., a^2 + c^2 = b^2 + d^2). Rhombi and kites are examples where this perpendicularity holds inherently.^[3] The taxonomy exhibits a clear hierarchy: a square is the intersection of a rectangle and a rhombus, combining equal sides with right angles, and thus also qualifies as a parallelogram. More broadly, parallelograms encompass rectangles, rhombi, and squares, while trapezoids stand apart due to their single parallel pair.^[62] A tangential quadrilateral allows an incircle tangent to all four sides, requiring the sums of the lengths of opposite sides to be equal (i.e., a + c = b + d). Rhombi are always tangential, and the area can be expressed as the inradius times the semiperimeter.^[29] A bicentric quadrilateral is both cyclic and tangential, possessing both an incircle and a circumcircle; it satisfies advanced relations between the inradius r, circumradius R, and side lengths, such as A = \sqrt{abcd} for its area. Squares are the most symmetric bicentric examples.^[64]

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