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Bisection

In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a bisector. The most often considered types of bisectors are the segment bisector, a line that passes through the midpoint of a given segment, and the angle bisector, a line that passes through the apex of an angle (that divides it into two equal angles). In three-dimensional space, bisection is usually done by a bisecting plane, also called the bisector.

Fundamental Bisectors

Line Segment Bisector

A perpendicular bisector of a line segment is a line that intersects the segment at its midpoint and forms a 90-degree angle with it, thereby dividing the segment into two equal parts.^[1] This line serves as the locus of all points in the plane that are equidistant from the two endpoints of the segment.^[2] Key geometric properties of the perpendicular bisector include its perpendicular orientation to the original segment at the exact midpoint, ensuring symmetry across the line. Any point on this bisector maintains equal distances to both endpoints, a fundamental characteristic that underpins its utility in geometric constructions.^[2]^[1] The perpendicular bisector can be constructed using a straightedge and compass through the following steps:

Place the compass point at one endpoint of the segment (say, A) and set the compass width to more than half the segment's length; draw two arcs, one above and one below the segment.
Without changing the compass width, place the point at the other endpoint (B) and draw two more arcs that intersect the previous arcs at two points (P and Q).
Use the straightedge to connect P and Q; this line intersects the segment at its midpoint and is perpendicular to it.^[3]^[1]

In coordinate geometry, for a line segment joining points (x_1, y_1) and (x_2, y_2), the perpendicular bisector's equation is derived by first finding the midpoint \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) and the slope of the original segment m = \frac{y_2 - y_1}{x_2 - x_1}. The slope of the bisector is then the negative reciprocal, m_p = -\frac{1}{m}, and the line equation is y - y_m = m_p (x - x_m), where (x_m, y_m) is the midpoint.^[4] In basic geometry, perpendicular bisectors are applied to locate the centers of circles passing through the two endpoints of the segment, as these centers lie on the bisector due to equal radii. They also facilitate symmetry constructions, such as reflecting points across the line to maintain equal distances.^[1]^[2]

Angle Bisector

An angle bisector is a ray, line, or line segment that divides a given angle into two congruent adjacent angles, each measuring half the measure of the original angle.^[5] This division ensures that the two resulting angles are equal in measure, forming a fundamental concept in plane geometry for achieving angular symmetry.^[5] A key property of the angle bisector is that every point on it is equidistant from the two sides of the original angle, where distance is measured perpendicularly to the sides.^[6] This equidistance theorem implies that the angle bisector serves as the locus of all points inside the angle that maintain equal perpendicular distances to its bounding rays.^[6] To construct an angle bisector using a straightedge and compass, place the compass point at the vertex of the angle and draw an arc that intersects both sides of the angle at equal distances from the vertex, creating two points of intersection.^[7] Then, with the compass point on each intersection point, draw two arcs inside the angle that intersect each other at a single point; finally, use the straightedge to draw a line from the vertex through this intersection point, which forms the angle bisector.^[7] In coordinate geometry, for an angle formed at the origin by two lines with equations a_1 x + b_1 y = 0 and a_2 x + b_2 y = 0, the equations of the angle bisectors can be derived by normalizing the line equations and setting their signed distances equal in magnitude.^[8] Specifically, the bisector equations are given by:

\frac{a_1 x + b_1 y}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2 x + b_2 y}{\sqrt{a_2^2 + b_2^2}}

The positive sign corresponds to one bisector, and the negative sign to the other, reflecting the two possible directions of equal angular division.^[8] This formulation arises from the condition that points on the bisectors are equidistant from the two lines.^[9] Angle bisectors find basic applications in geometry for dividing polygons into symmetric regions, such as halving central angles in regular polygons to locate vertices or create balanced subdivisions.^[10] They are also essential in constructing symmetric figures, like equilateral triangles from given angles, by ensuring precise angular equality across multiple components.^[10]

Bisectors in Triangles

Angle Bisectors

In a triangle, the three angle bisectors are concurrent at a single point known as the incenter, which serves as the center of the incircle tangent to all three sides.^[11] This concurrency point is equidistant from all sides, with the distance equal to the inradius. The incenter's location ensures that the incircle touches each side at exactly one point. Connecting the incenter to the vertices divides the triangle into three smaller triangles whose areas are \frac{1}{2} r times the respective side lengths, where r is the inradius.^[11] The angle bisector theorem states that the bisector of an angle in a triangle divides the opposite side into two segments proportional to the adjacent sides. Specifically, in triangle ABC with angle bisector from vertex A meeting side BC at D, the ratio BD/DC = AB/AC.^[12] One proof relies on similar triangles: drawing a line parallel to BC from a point on the bisector creates two triangles similar to the original by AA similarity (sharing angles and equal bisected halves), leading to the side ratio equality.^[13] An alternative proof uses area ratios, noting that the areas of triangles ABD and ACD share the same height from A, so their base ratio BD/DC equals the side ratio AB/AC via equal areas per unit base.^[13] The length of an angle bisector from vertex A to the opposite side a (with adjacent sides b and c) is given by

t_a = \frac{2bc}{b+c} \cos\left(\frac{A}{2}\right),

or equivalently, via a variant of Stewart's theorem,

t_a^2 = bc \left[1 - \left(\frac{a}{b+c}\right)^2\right].

^[14] These formulas derive from applying the law of cosines in the sub-triangles formed by the bisector and resolving the resulting expressions. For applications, the incenter's coordinates in a triangle with vertices (x_1, y_1), (x_2, y_2), (x_3, y_3) and opposite sides a, b, c are

\left( \frac{a x_1 + b x_2 + c x_3}{a + b + c}, \frac{a y_1 + b y_2 + c y_3}{a + b + c} \right),

facilitating computations in coordinate geometry and emphasizing the weighted average based on side lengths.^[15] This tangential property underscores the incenter's role as the point where perpendicular distances to the sides equal the inradius, enabling constructions of incircles and ex circles.^[11] Triangles with integer side lengths and integer angle bisector lengths, often Heronian triangles (integer sides and area), satisfy conditions where the bisector formula yields an integer under the square root. For instance, the isosceles triangle with sides 5, 5, 6 has an angle bisector length of 4 from the vertex opposite the base of 6, as t^2 = 5 \cdot 5 \left[1 - (6/10)^2\right] = 16.^[16] Integrality requires that bc \left[1 - (a/(b+c))^2\right] is a perfect square, a Diophantine condition explored in studies of rational bisectors, with such triangles forming a countable set generated parametrically similar to Heronian parametrizations.^[17]

Side and Median Bisectors

In a triangle, a median is a line segment that connects a vertex to the midpoint of the opposite side, thereby bisecting that side into two equal segments.^[18] The three medians of any triangle are concurrent, intersecting at a single point known as the centroid, which serves as the balance point or center of mass of the triangle.^[19] The centroid divides each median in a 2:1 ratio, with the longer segment (twice the length of the shorter one) extending from the vertex to the centroid.^[20] The length of a median from vertex A opposite side a (with adjacent sides b and c) is given by the formula

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

derived from the Apollonius theorem relating the medians to the sides.^[18] This formula allows computation of median lengths solely from the triangle's side lengths, facilitating analysis in coordinate or vector-based geometries without explicit coordinates.^[21] The perpendicular bisector of a triangle's side is the line that passes through the midpoint of the side and is perpendicular to it, forming the locus of all points equidistant from the endpoints of that side.^[22] In any triangle, the three perpendicular bisectors are concurrent at the circumcenter, the center of the unique circle (circumcircle) that passes through all three vertices.^[23] The circumradius R, distance from the circumcenter to any vertex, underscores the perpendicular bisectors' role in defining the triangle's circumcircle.^[24] Medians and perpendicular bisectors contribute to key collinearities in triangle geometry via the Euler line, which passes through the centroid (intersection of medians), the circumcenter (intersection of perpendicular bisectors), and the orthocenter (intersection of altitudes).^[24] In general triangles, the centroid divides the segment joining the orthocenter and circumcenter in a 2:1 ratio, with the longer part toward the orthocenter; this relation holds universally but manifests distinctly in special cases, such as equilateral triangles where all three centers coincide.^[25] For isosceles or right triangles, the Euler line aligns with a median or altitude, simplifying computations.^[24] These bisectors find applications in identifying triangle centers and in vector geometry, where the centroid's position vector is the average of the vertices' position vectors, enabling barycentric coordinates and mass distribution models.^[26] Perpendicular bisectors, meanwhile, support constructions for equidistant points, such as in Voronoi diagrams or optimization problems involving triangle symmetries.^[22]

Area and Perimeter Bisectors

In a triangle, an area bisector is any straight line that intersects the boundary at two points and divides the interior into two regions of equal area, each half the total area. There are infinitely many such lines, unlike the unique angle bisector from each vertex. The medians serve as special area bisectors; each connects a vertex to the midpoint of the opposite side and divides the triangle into two smaller triangles of equal area, since the sub-triangles share the same height from the vertex and have bases half the length of the opposite side. The three medians intersect concurrently at the centroid, the triangle's center of mass, which divides each median in a 2:1 ratio with the longer portion toward the vertex.^[27] From a given vertex, the unique area bisector is the median, which strikes the midpoint of the opposite side. The length of this median from vertex A (opposite side a) to the midpoint of side a is given by

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

where b and c are the lengths of the other two sides. In an equilateral triangle with side length a, all medians have length \frac{\sqrt{3}}{2} a and coincide with the altitudes, angle bisectors, and perpendicular bisectors. In a scalene triangle, such as one with sides 3, 4, 5, the medians bisect the area but vary in length (e.g., the median to the side of length 5 is \frac{1}{2} \sqrt{2 \cdot 3^2 + 2 \cdot 4^2 - 5^2} = \frac{5}{2} = 2.5) and do not align with other bisectors. The envelope of all area bisectors forms a deltoid (a hypocycloid with three cusps) inside the triangle, with points inside the deltoid lying on three area bisectors and points outside on one; this deltoid has area approximately 0.01986 times that of the triangle. Alternatively, area bisectors are tangent to three hyperbolas whose asymptotes align with the triangle's sides.^[27]^[28] Area bisectors appear in optimization and dissection problems, such as partitioning a triangle into equal-area regions or analyzing mass distribution in engineering designs. Medians, as a subset, provide a natural reference for such divisions.^[29] A perimeter bisector is any straight line that intersects the triangle's boundary at two points, dividing the perimeter into two arcs of equal length, each equal to the semiperimeter s = (a + b + c)/2. Like area bisectors, there are infinitely many perimeter bisectors, and their envelope consists of three parabolic arcs connecting the midpoints of the sides. The cevians called splitters are specific perimeter bisectors from the vertices; the splitter from vertex A intersects side BC (length a) at point D, where BD = s - c and CD = s - b (with b and c the lengths opposite B and C). The three splitters concur at the Nagel point, a triangle center analogous to the incenter but associated with excircles and perimeter properties.^[30]^[31]^[32] In an equilateral triangle, the splitters coincide with the medians and angle bisectors, each dividing the opposite side at its midpoint (s - a = a/2). In a scalene triangle, such as the 3-4-5 triangle (s = 6), the splitter from the vertex opposite the side of length 5 divides that side into segments of length 6 - 3 = 3 and 6 - 4 = 2, resulting in distinct lines that meet at the Nagel point. Perimeter bisectors relate to the semiperimeter in constructions and have applications in fair division problems, polygon partitioning, and studying triangle centers; notably, lines that simultaneously bisect both area and perimeter must pass through the incenter.^[31]^[33]

Bisectors in Quadrilaterals

Angle and Side Bisectors

In a quadrilateral, the four interior angle bisectors intersect the adjacent bisectors at four points that lie on a common circle.^[34] The angle bisectors are concurrent if and only if the quadrilateral is tangential, in which case they meet at the incenter, the center of the incircle tangent to all four sides.^[35] This concurrency property extends the analogous situation in triangles, where angle bisectors always concur at the incenter.^[35] Special cases exhibit notable intersection properties. In a rhombus, each diagonal serves as the bisector of the angles at the vertices it connects, due to the equal side lengths creating congruent triangles along the diagonals.^[36] For an ex-tangential quadrilateral, which admits an excircle tangent to the extensions of all four sides, the internal angle bisectors at two opposite vertices and the external angle bisectors at the other two vertices concur at the excenter.^[37] These concurrency conditions highlight symmetry in certain quadrilaterals. In a kite, the axis of symmetry along the diagonal between the equal adjacent sides acts as the angle bisector for the vertex angles at the ends of that diagonal.^[38] Similarly, in a parallelogram, the angle bisectors concur only if it is a rhombus, where the diagonals provide the bisections.^[36] The perpendicular bisectors of the sides of a quadrilateral, which divide each side into two equal segments at right angles, are concurrent if and only if the quadrilateral is cyclic, meeting at the circumcenter equidistant from all vertices.^[39] In tangential quadrilaterals, these perpendicular bisectors form another quadrilateral whose properties relate indirectly to the incircle through constructions involving the points of tangency, though they do not generally pass through the incenter.^[40]

Diagonal Bisectors

In parallelograms, the diagonals bisect each other, intersecting at their respective midpoints to divide the figure into two pairs of congruent triangles. This midpoint intersection occurs regardless of the specific angles or side lengths, as long as opposite sides are parallel and equal. The property holds because the vector sum of adjacent sides \vec{AB} + \vec{AD} forms one diagonal, while \vec{CB} + \vec{CD} forms the other, and their midpoints coincide at the average position \frac{1}{2}(\vec{AB} + \vec{AD}).^[41]^[42] In a rhombus, which is a parallelogram with all sides equal, the diagonals not only bisect each other at the midpoint but also do so perpendicularly, forming right angles at the intersection and dividing the rhombus into four right-angled triangles. This perpendicularity arises from the equal side lengths, ensuring the diagonals act as perpendicular bisectors.^[43] For general quadrilaterals, the concept extends to bisect-diagonal quadrilaterals, where at least one diagonal bisects the other into two equal segments at their intersection point. Examples include kites, where one diagonal bisects the other, and parallelograms, where mutual bisection occurs. A key property is that the bisecting diagonal also divides the quadrilateral's area into two equal parts. Additionally, the bimedians (lines joining midpoints of opposite sides) intersect on the non-bisecting diagonal if and only if the quadrilateral is bisect-diagonal.^[44] The perpendicular bisectors of the diagonals in a general quadrilateral intersect at a point that relates to symmetry properties, though this intersection does not always yield a special center unless the figure is cyclic or orthodiagonal. In special cases, such as a square, one diagonal serves as the perpendicular bisector of the other, with their intersection at 90 degrees and equal lengths. For broader analysis, Varignon's theorem connects midpoint properties: the quadrilateral formed by joining the midpoints of the sides of any quadrilateral is a parallelogram, whose diagonals bisect each other and whose sides are parallel to the bimedians of the original figure, linking directly to diagonal midpoint considerations. The area of this Varignon parallelogram is half the area of the original quadrilateral.^[45]^[46]^[44] Length formulas for bisecting diagonals provide quantitative insights. In a parallelogram with adjacent sides a and b and included angle \theta, the diagonals d_1 and d_2 (which bisect each other) are:

d_1 = \sqrt{a^2 + b^2 + 2ab \cos \theta}, \quad d_2 = \sqrt{a^2 + b^2 - 2ab \cos \theta}.

^[47] These properties find applications in vector-based proofs of quadrilateral theorems, such as demonstrating that the vector midpoint equality implies parallelism in Varignon's construction or confirming area bisection without coordinate geometry. Such proofs underpin classifications of quadrilaterals and optimizations in computational geometry.^[42]^[44]

Area Bisectors

In parallelograms, the diagonals each bisect the area of the figure by dividing it into two congruent triangles of equal area. Furthermore, the intersection point of the diagonals serves as the center such that any straight line passing through this point divides the parallelogram into two regions of equal area, a property that characterizes parallelograms among quadrilaterals.^[48] For general quadrilaterals, multiple straight lines can bisect the area, and their construction typically involves coordinate geometry methods, where the quadrilateral is placed on a coordinate plane and the equation of the line is solved such that the areas of the resulting polygons on either side equal half the total area. The position of such a bisector relative to the vertices often divides the figure in ratios proportional to the sub-areas formed by the intersecting line, ensuring balance through integration or summation of triangular components. In isosceles trapezoids, the line joining the midpoints of the two parallel bases bisects the area due to symmetry. In general trapezoids, this line passes through the intersection point of the diagonals.^[49] In kites, the diagonal connecting the two vertices with equal adjacent sides acts as the axis of symmetry and bisects the area, dividing the kite into two congruent right triangles of equal area. This diagonal also perpendicularly bisects the other diagonal, reinforcing the equal division.^[50] These area bisectors find applications in computational geometry for tasks such as polygon partitioning and fair division algorithms, where equal-area splits are required for optimization in design and simulation.^[50]

Bisectors in Other Plane Figures

Circles and Ellipses

In a circle of radius r, any diameter serves as an area bisector, dividing the disk into two regions of equal area \frac{1}{2} \pi r^2. This follows from the central symmetry of the circle, where every line through the center maps one side to the other via 180-degree rotation, ensuring equal areas. Similarly, each diameter bisects the perimeter (circumference), splitting the boundary into two equal arcs of length \pi r. No chord displaced from the center can bisect the area, as the smaller segment it forms has area strictly less than half the total, given by the formula r^2 \cos^{-1}\left(\frac{d}{r}\right) - d \sqrt{r^2 - d^2} for distance d > 0 from the center. Ellipses share this central symmetry with circles but exhibit directional variation due to differing semi-major axis a and semi-minor axis b (with a > b). Consequently, every line through the center bisects both the enclosed area \pi a b and the perimeter, approximately $4a E(e) where E(e) is the complete elliptic integral of the second kind and e = \sqrt{1 - (b/a)^2} is the eccentricity. This property—that all area bisectors (and separately, all perimeter bisectors) concur at a single point—characterizes central symmetry among planar convex sets. The major and minor axes exemplify such bisectors, leveraging the ellipse's reflectional symmetry along these axes. A general bisector through the center at angle \theta to the major axis has parametric equations

x = t \cos \theta, \quad y = t \sin \theta.

Substituting into the ellipse equation \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 yields the intersection parameter

t = \pm \left( \frac{\cos^2 \theta}{a^2} + \frac{\sin^2 \theta}{b^2} \right)^{-1/2}.

The chord length is thus

$2 \left( \frac{\cos^2 \theta}{a^2} + \frac{\sin^2 \theta}{b^2} \right)^{-1/2} = \frac{2ab}{\sqrt{b^2 \cos^2 \theta + a^2 \sin^2 \theta}},

highlighting the role of eccentricity in scaling the bisector length with orientation. These bisector properties underpin applications in orbital mechanics, where elliptical planetary orbits enable symmetric area divisions through the geometric center for trajectory analysis, and in optical lens design, where elliptical surfaces exploit central symmetry for aberration minimization and uniform light distribution.

Parabolas

The axis of a parabola functions as its fundamental symmetry bisector, partitioning the curve into two congruent mirror images on either side. For the standard equation y^2 = 4ax, the axis coincides with the x-axis, extending through the vertex at the origin and the focus at (a, 0), while remaining perpendicular to the directrix x = -a. This symmetry ensures that corresponding points on opposite sides of the axis are equidistant from both the focus and the directrix, embodying the geometric definition of the parabola.^[51]^[52] Focal chords, defined as line segments joining two points on the parabola and passing through the focus, exhibit distinct geometric properties that facilitate analysis of bisectors. In parametric form, points on y^2 = 4ax are represented as (at^2, 2at); a focal chord connects parameters t_1 and t_2 satisfying t_1 t_2 = -1. The midpoint of such a chord is located at \left( a \frac{t_1^2 + t_2^2}{2}, a(t_1 + t_2) \right), and the slope of the chord itself is \frac{2}{t_1 + t_2}.^[53] The perpendicular bisector of a focal chord passes through this midpoint and has a slope equal to the negative reciprocal of the chord's slope, namely -\frac{t_1 + t_2}{2}. More generally, for any chord of the parabola with midpoint (x_1, y_1), the equation of the chord is y y_1 = 2a (x + x_1), implying a slope of \frac{2a}{y_1}; thus, the perpendicular bisector has slope -\frac{y_1}{2a} and the equation

y - y_1 = -\frac{y_1}{2a} (x - x_1).

These constructions highlight the interplay between chord midpoints and the parabola's curvature, with the perpendicular bisector providing a line of symmetry local to the chord segment. For focal chords specifically, the parameters t_1 and t_2 ensure the midpoint lies outside the directrix, influencing the bisector's orientation relative to the axis.^[54] Area bisectors under a parabola refer to lines that divide the region bounded by the curve, a chord, and the axis into two equal areas, often applied to parabolic segments. The area of a parabolic segment formed by a chord of base b and height h is \frac{2}{3} b h, derived from integration or geometric exhaustion methods. To find an area bisector, such as a vertical line dividing the segment equally, one solves for the position where the integral of the parabolic function yields half the total area; for instance, under y = \frac{x^2}{4p} from x = -c to x = c, the axis itself (x = 0) serves as the bisector due to symmetry, but tilted lines require solving transcendental equations for equal segmental areas. These bisectors are useful in optimization problems involving parabolic arches or trajectories.^[55] In practical applications, parabolic bisectors underpin designs in physics and engineering. In projectile motion, the parabolic trajectory of an object under gravity has its axis of symmetry bisecting the path at the vertex, corresponding to maximum height, which aids in predicting range and apex for ballistic calculations. Similarly, in reflector design, the axis as symmetry bisector ensures that incident rays parallel to it converge at the focus after reflection off the parabolic surface, a property exploited in satellite dishes, solar concentrators, and optical instruments for efficient signal focusing. This reflective symmetry parallels the major axis role in elliptical mirrors but emphasizes the open-ended nature of parabolas for unbounded ray collection.^[56]/05:_Conic_Sections__Circle_and_Parabola/5.03:_Applications_of_the_Parabola)

General Polygons

In general polygons with n > 4 sides, side bisectors refer to the perpendicular bisectors of each side, which are lines passing through the midpoint of a side and perpendicular to it, serving as the locus of points equidistant from the side's endpoints. These bisectors form an arrangement of n lines whose intersections can identify potential circumcenters for subsets of vertices, though in non-cyclic polygons, they generally do not concur at a single point. The envelope of these perpendicular bisectors, studied in projective geometry, is constructed via projective normals to the sides, yielding a new polygon whose vertices lie at intersections of consecutive bisectors; this mapping preserves projective equivalence and exhibits dynamical properties under iteration, such as conjugation to linear maps on elliptic curves for pentagons.^[57] Angle bisectors in general polygons extend the interior angle divisions seen in lower-sided cases, but their behavior differs markedly between regular and irregular forms. In regular n-gons, the angle bisectors from each vertex concur at the center due to rotational symmetry, coinciding with the lines from the center to the vertices. This concurrency simplifies geometric constructions, such as locating the incenter, which aligns with the circumcenter and centroid. In irregular polygons, angle bisectors rarely concur, lacking a unified center, and their arrangement depends on varying interior angles, reducing symmetry and complicating intersections compared to the unified point in regular cases. Area bisectors are lines that divide a polygon into two regions of equal area, with algorithms enabling computation for arbitrary directions. A linear-time algorithm trapezoidizes the polygon and applies a prune-and-search technique to find such a line in a specified direction, achieving O(n) complexity for simple polygons with n vertices. Simple polygons can have up to \Theta(n^2) combinatorially distinct area bisectors, matching the upper bound from arrangements of dual lines. For example, in a regular pentagon, any line through the center bisects the area, as the equal central angles ensure each half-plane covers exactly 180 degrees of the total angular span, proportionally halving the triangular sectors' areas; similarly, regular hexagons exhibit this property due to their six equal 60-degree sectors. In irregular polygons, symmetry is absent, leading to more varied bisector configurations and requiring computational methods for precise division. Perimeter bisectors are lines that split the boundary length into two equal parts, with exactly one such line existing per direction \theta in convex polygons. These bisectors form a continuous envelope consisting of parabolic segments and isolated points, continuous only in special cases like equilateral triangles or parallelograms. In regular hexagons, symmetry lines (e.g., those connecting opposite vertices) serve as perimeter bisectors, aligning with the equal side lengths; for pentagons, rotational symmetry yields five principal bisectors along radial lines, but general directions require solving for the unique offset that halves the perimeter. Irregular polygons lack these symmetries, resulting in envelopes with cusps and requiring algorithmic tracing of hyperbolic arcs asymptotic to side pairs. The distinction between regular and irregular polygons lies in symmetry reductions: regular forms exhibit concurrent bisectors and radial simplicity, minimizing distinct lines (e.g., all passing through the center), while irregular ones demand full computational enumeration, with envelopes showing greater complexity and up to quadratic variety in bisectors. These properties find applications in computer graphics, where medial axes—comprising segments of perpendicular bisectors between boundary elements—enable skeletonization for shape deformation and thinning algorithms, and in tiling designs, where bisectors guide symmetric subdivisions for periodic patterns using regular polygons.^[58]^[59]^[32]^[60]^[61]

Spatial Bisectors

Line Segment Bisectors in Space

In three-dimensional space, the perpendicular bisector of a line segment with endpoints A and B is the unique plane that passes through the midpoint of the segment and is perpendicular to the direction of the segment, forming the locus of all points equidistant from A and B.^[62] This extends the two-dimensional concept, where the bisector is a line, to a plane that divides space into two half-spaces—one containing points closer to A and the other closer to B. To construct the bisecting plane using coordinates, let the endpoints be A = (x_1, y_1, z_1) and B = (x_2, y_2, z_2). The midpoint M is calculated as

M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right).

The direction vector of the segment is \vec{d} = (x_2 - x_1, y_2 - y_1, z_2 - z_1), which serves as the normal vector to the plane. The equation of the plane is then

\vec{d} \cdot (\vec{X} - M) = 0,

where \vec{X} = (x, y, z) is any point on the plane.^[62] Using vector methods, this can also be derived from the condition |\vec{X} - A| = |\vec{X} - B|, which simplifies to the same plane equation after squaring and rearranging terms.^[62] The plane is infinite in extent and unique for the given segment, containing all points where the distance to A equals the distance to B, and it is perpendicular to the segment at the midpoint.^[62] In applications, such planes play a key role in three-dimensional symmetry analysis; for instance, in molecular geometry, a mirror plane often acts as the perpendicular bisector of a bond between two identical atoms, ensuring the molecule's symmetry under reflection. Similarly, in crystal lattices, these planes are used to define the boundaries of the Wigner-Seitz cell by connecting perpendicular bisectors between a lattice point and its nearest neighbors, providing a primitive representation of the lattice structure.^[63]

Volume Bisectors

A volume bisector in three-dimensional geometry is a plane that partitions a solid object into two regions each possessing half the total volume of the solid.^[64] This notion extends the concept of area bisectors observed in two-dimensional shapes, where lines divide regions of equal area. For highly symmetric solids like the sphere, any plane passing through the geometric center serves as a volume bisector, a consequence of the sphere's isotropic symmetry ensuring uniform volume distribution on either side. Similarly, in a cube, central symmetry guarantees that every plane through the centroid bisects the volume exactly, regardless of orientation. In polyhedra such as the regular tetrahedron, volume bisectors include any plane containing a bimedian (the line segment joining the midpoints of two opposite edges); there are three such bimedians, each giving a family of bisecting planes that divide the solid into two regions of equal volume. In general polyhedra, volume bisectors often intersect the centroid, which coincides with the center of mass for uniform density solids, but not all such planes pass through this point unless the polyhedron exhibits central symmetry. Non-symmetric polyhedra admit infinitely many volume bisectors, forming a continuous family whose directions can be parameterized; for instance, in a tetrahedron, these planes envelope a curved surface known as the bisectrix.^[65] The multiplicity of bisectors—meaning the dimensionality of the set of such planes—is typically two (a continuum of orientations) for asymmetric solids, contrasting with the infinite but symmetrically constrained cases in centrally symmetric ones.^[64] Applications of volume bisectors appear in computational geometry for partitioning 3D models, such as in 3D printing where bisecting planes facilitate splitting objects into printable segments of equal material volume to optimize fabrication. In physics, they model equitable division of fluid volumes in containers, aiding simulations of balanced partitioning under gravity or pressure. Resource allocation problems in engineering also employ these bisectors to divide spatial domains fairly, as in distributing storage volumes in 3D warehouses. In higher dimensions, the concept generalizes to hyperplanes that bisect the content (n-dimensional volume) of an n-dimensional body; the ham-sandwich theorem ensures the existence of such hyperplanes for multiple measures simultaneously, with central symmetry again implying that all hyperplanes through the center perform the bisection.^[64]

References

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3.03: Bisection Methods for Solving a Nonlinear Equation
Oct 5, 2023 · One of the first numerical methods developed to find the root of a nonlinear equation f ⁡ ( x ) = 0 was the bisection method (also called the ...Lesson 2: Bisection Method... · Bisection method · Lesson 3: Application of...Missing: definition | Show results with:definition
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Bisection Method - an overview | ScienceDirect Topics
The Bisection Method is a simple and reliable numerical method used to calculate the roots of a function within a given interval.
[3]
Perpendicular Bisector - Definition, Construction, Properties, Examples
Perpendicular bisector is constructed using a straight ruler and a compass using the following steps: Draw a line segment AB of any length with a straight edge ...
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Perpendicular Bisectors | CK-12 Foundation
Nov 1, 2025 · A perpendicular bisector is a line that intersects a line segment at its midpoint and is perpendicular to that line segment.
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Perpendicular bisector of a line segment - Math Open Reference
This construction shows how to draw the perpendicular bisector of a given line segment with compass and straightedge or ruler.
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Equation of a Perpendicular Bisector - Math - StudySmarter
Jan 27, 2022 · For instance, you can find the midpoint of the segment of the line with the endpoints (a, b) and (c, d) through the formula: ( a + c 2 , b + d 2 ) ...
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Angle Bisector -- from Wolfram MathWorld
The (interior) bisector of an angle, also called the internal angle bisector (Kimberling 1998, pp. 11-12), is the line or line segment that divides the angle ...
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Construction of the Angle Bisector
The angle bisector of an angle is the locus of points equidistant from the sides of the angle. Of course the notion of angle needs to be defined accurately.
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Angle Bisector Construction - Math is Fun
How to construct an Angle Bisector (halve the angle) using just a compass and a straightedge.
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Equations of the Bisectors of the Angles between Two Straight Lines
a1x+b1y+c1√a21+b21 = + a2x+b2y+c2√a22+b22, which is the required bisector of the angle containing the origin. Note: The bisector of the angle containing the ...<|separator|>
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Angle Bisectors - Study Material for IIT JEE - askIITians
When both c1 and c2 are of the same sign, the equation of the bisector of the angle which contains the origin is (a1x + b1y + c1)/√(a12 + b12) = + (a2x + b2y + ...What is an Angle Bisector? · How can we differentiate...
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Angle bisector - BYJU'S
Angle bisector in geometry refers to a line that splits an angle into two equal angles. Bisector means the thing that bisects a shape or an object into two ...
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[PDF] Chapter 4 - Concurrency of Lines in a Triangle
Theorem 4.7 The angle bisectors of a triangle intersect at a common point I called the incenter, which is the center of the unique circle inscribed in the ...
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Euclid's Elements, Book VI, Proposition 3 - Clark University
This proposition characterizes an angle bisector of an angle in a triangle as the line that partitions the base into parts proportional to the adjacent sides.Missing: similar | Show results with:similar
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Angle Bisector Theorem - MathBitsNotebook(Geo)
The "Angle Bisector" Theorem says that an angle bisector of a triangle will divide the opposite side into two segments that are proportional to the other two ...
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[PDF] Cevians - People | MIT CSAIL
The angle bisectors of a triangle intersect at the incenter, the center ... Angle Bisector Theorem: a m b n. = Length of an Angle Bisector: t ab c. a b c ...
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Incenter -- from Wolfram MathWorld
(x_3,y_3) , the Cartesian coordinates of the incenter are given by. (x_I,y_I)=((ax_1+bx_2+cx_3)/(. (2). The distance between the incenter and circumcenter is ...
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Triangle Solver - Dr Ron Knott
There is an integer-sided triangle with one 60° angle: 5,7,8 and another with a 120° angle (so it has an exterior angle of 60°): 3,7,8. There are non-integer ...
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On triangles with rational altitudes, angle bisectors or medians
May 9, 2025 · This thesis is principally concerned with integer-sided triangles. Such a triangle is called a Heron triangle if its area is also an integer.
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Length of a Median - Interactive Mathematics Miscellany and Puzzles
The length of the median m_a from A is given by \displaystyle m_{a}^{2}=\frac{b^{2}+c^{2}}{2}-\frac{a^2}{4}, or \displaystyle m_{a}Missing: definition | Show results with:definition
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Triangle Centroid -- from Wolfram MathWorld
The triangle centroid is the geometric center of mass, the intersection of the three medians, and is located 2/3 of the way from each vertex to the opposite ...<|control11|><|separator|>
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Centroid - MathBitsNotebook (Geo)
The centroid divides the medians into a 2:1 ratio. The portion of the median nearest the vertex is twice as long as the portion connected to the midpoint of the ...<|control11|><|separator|>
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https://www.varsitytutors.com/hotmath/hotmath_help/topics/circumcenter-theorem
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Triangle Circumcenter definition - Math Open Reference
The circumcenter is the point where the perpendicular bisectors of a triangle intersect. The circumcenter is also the center of the triangle's circumcircle.
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Circumcenter Theorem - Varsity Tutors
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is equidistant from the vertices of the triangle.
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Euler Line | Brilliant Math & Science Wiki
The Euler line of a triangle is a line going through several important triangle ... medians, green perpendicular bisectors, and the red line is the Euler line.
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Euler line - Math Open Reference
The three perpendicular bisectors (here colored magenta) are the lines that cross each side of the triangle at right angles exactly at their midpoint. See ...
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Section Formula and Centroid of a Triangle Using Vectors | CK-12 ...
Aug 26, 2025 · The centroid G is the point where the medians of a triangle intersect and is found using: · Formula: g → = u → + v → + w → 3 · Each median joins a ...
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Medians and area bisectors of triangles
The intersection of the deltoid and a median or other area bisector is 1/sqrt(2) - 1/2 = 0.2071... of the length of that median or area bisector. Any ...
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Triangle Area Bisectors - Wolfram Demonstrations Project
Area bisectors are straight lines that divide a triangle into two parts of equal area. They are all tangent lines to three different hyperbolas.
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Triangle slicing - Bluffton University
... triangle? Details: Given △ABC, an area-bisector is a segment with endpoints on the triangle, which divides the triangle in half (by area). The triangle's ...
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Splitter -- from Wolfram MathWorld
A perimeter-bisecting line segment which originates at a vertex of a polygon. The three splitters of a triangle concur in a point known as the Nagel point Na.
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Nagel Point -- from Wolfram MathWorld
The Nagel point (Na) is where lines A_1T_1, A_2T_2, and A_3T_3 concur, and is also known as the bisected perimeter point.
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[PDF] PERIMETER BISECTORS, CUSPS, AND KITES
This envelope is continuous through the perimeter bisector that is also the triangle's line of symmetry. Definition 2.4. We define the mouth of a parabola to be ...
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[PDF] lines simultaneously bisecting the perimeter and area of a triangle
area bisector in angle A. A perimeter-area bisector YZ in angle A of triangle ABC depends on the lengths of AY = y and AZ = z. We require y + z = s and yz ...
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Quadrilateral -- from Wolfram MathWorld
(Honsberger 1995, pp. 39-40). The four angle bisectors of a quadrilateral intersect adjacent bisectors in four concyclic points (Honsberger 1995, p. 35).Missing: properties | Show results with:properties
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Tangential Quadrilateral | Brilliant Math & Science Wiki
A circle can be inscribed in a quadrilateral if and only if the angle bisectors of the four angles of the quadrilateral are concurrent. Join Brilliant The ...
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Lesson Diagonals of a rhombus bisect its angles
In a rhombus, the diagonals are the angle bisectors. 2. If in a parallelogram the two diagonals are the angle bisectors, then the parallelogram is a rhombus.
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[PDF] CHARACTERIZATIONS OF EXTANGENTIAL QUADRILATERALS
An extangential quadrilateral is a convex quadrilateral with an excircle,. i.e. an external circle tangent to the extensions of all four sides, see Figure. 1. A ...
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Classification of Quadrilaterals
You can display its diagonals, angle bisectors and the perpendicular bisectors of its sides. ... Here is a list of all the properties of quadrilaterals ...
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https://www.mathopenref.com/parallelogramdiags.html
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Quadrilaterals Formed by Perpendicular Bisectors
The perpendicular bisectors of the sides of a quadrilateral ABCD form a quadrilateral A1B1C1D1, and the perpendicular bisectors of the sides of A1B1C1D1 ...
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diagonals of a parallelogram bisect each other - Math Open Reference
In any parallelogram, the diagonals (lines linking opposite corners) bisect each other. That is, each diagonal cuts the other into two equal parts.
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Diagonals of a parallelogram - Richard Fitzpatrick
The use of vectors is very well illustrated by the following rather famous proof that the diagonals of a parallelogram mutually bisect one another.
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Proof: Rhombus diagonals are perpendicular bisectors (video)
Aug 9, 2016 · Sal proves that the diagonals of a rhombus are perpendicular, and that they intersect at the midpoints of both.
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[PDF] Properties of bisect-diagonal quadrilaterals
Aug 4, 2020 · Theorem 3: The bimedians in a convex quadrilateral intersect on a diagonal if, and only if, it is a bisect-diagonal quadrilateral. Proof: A ...
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Varignon's Theorem -- from Wolfram MathWorld
The area of the Varignon parallelogram of a convex quadrilateral is half that of the quadrilateral, and the perimeter is equal to the sum of the diagonals of ...
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[PDF] Exploring Diagonals of Quadrilaterals - Summary of lesson
o If in a quadrilateral one diagonal is the perpendicular bisector of the other diagonal, then the quadrilateral must be a square. •. You might use Always ...
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Diagonal of Parallelogram - Formula, Properties, Examples - Cuemath
A simple formula which is used to find the length of the diagonals of a parallelogram needs the value of the interior angles and the length of the sides. For ...
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[PDF] Formulas for Diagonals of any Quadrilateral - AwesomeMath
We provide formulas for representing a diagonal of any cyclic quadrilateral, as well as a formula of representing a diagonal of any quadrilateral.
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[PDF] CHARACTERIZATIONS OF PARALLELOGRAMS PART 2
P. (e) there is a point X inside ABCD such that any line through X bisects the area of ABCD. (f) TABY + TCDY = TBCY + TDAY for any interior point Y. (g) the ...
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[PDF] [Kiselev,Givental]Kiselev's_Geom1.pdf - Cimat
a trapezoid. The parallel sides (AD and BC, Figure 103) of a trape- zoid are ... bisects the area of the quadrilateral. Computation problems. 522. In a ...
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[PDF] Equipartitioning and Balancing Points of Polygons - Semantic Scholar
Then it is an easy matter to see that the same diagonal bisects the area as well. ... (In this case ABCD is a kite). This completes the proof. Figure 6.
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Parabola -- from Wolfram MathWorld
### Summary of Parabola Properties
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Lesson 13: Graphs of Quadratic Functions & Vertex of a Parabola
The axis of symmetry is the vertical line x = h passing through the vertex, where h is the x -coordinate of the vertex. ☆ In general, how many intersection ...
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Focal chord of Parabola - Study Material for IIT JEE - askIITians
The extremities of a focal chord of the parabola y2 = 4ax may be taken as the points t and –1/t. Length of the chord. The abscissae of the points common to the ...
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[PDF] THE LOCUS OF THE KID-POINTS OF A SYSTEM OF CHORDS OF ...
y =- y, - - 4ax; . The mid-point (x z, y7) of this chord is x - y.2' - 2axz v_ v ... y') of the chord, is then: xZ— v'_ a'^y a*y, f b*xz* ) t. /. These are ...
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Parabolic Segment -- from Wolfram MathWorld
The arc length of a parabolic segment is given by s = 2int_0^asqrt(1+y^('2))dx, and the area is given by A = 4/3ah.
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Applied Use of Parabolas - MathBitsNotebook(A2)
Parabolas are used in parabolic dishes, radio antennas, flashlights, and the trajectory of objects under gravity. They are also used in suspension and arch ...
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[2202.09828] On projective evolutes of polygons - arXiv
Feb 20, 2022 · In this note we consider a projectively natural discrete analog of this construction: we define projective perpendicular bisectors of the sides ...
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A linear algorithm for bisecting a polygon - ScienceDirect.com
Mar 6, 1992 · This short note describes a linear-time algorithm to find a line in a given direction which bisects a simple polygon.
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[PDF] The Area Bisectors of a Polygon and Force Equilibria in ... - Research
For any given direction there is a unique area bisector. We denote the oriented bisector of P that makes an angle with the positive x-axis by B( ), and (because ...
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Lesson Regular polygons
It implies that all perpendicular bisectors of the sides of a regular polygon intersect in one point, or, in other words, are concurrent.<|control11|><|separator|>
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Medial Axis - an overview | ScienceDirect Topics
The medial axis M(C) of C is a tree whose edges are pieces of angular bisectors of C's sides. In fact, M(C) coincides with the part of the straight skeleton ...
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Equation of a plane equidistant from two points $A$ and $B
Aug 29, 2012 · Assuming you mean that all points in the plane are equidistant from A and B, then notice that the vector v:=v2−v1=(x2−x1,y2−y1,z2−z1)T.Equation of plane equidistant from 2 3d lines - Math Stack ExchangeFind equation for all points equidistant to two pointsMore results from math.stackexchange.com
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[PDF] CHAPTER 2 SOLID-STATE CHEMISTRY - NET
Is there a mirror plane to the molecular axis (σh)?. Are there n 2-fold ... Perpendicular bisector lines are then drawn to each of these vectors; the enclosed ...
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[PDF] 1 Crystal structures Masatsugu Sei Suzuki Department of Physics ...
Jan 22, 2019 · The dotted line denotes the perpendicular bisector. ... There is automatically a second mirror plane normal to the first mirror plane.
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Ham Sandwich Theorem -- from Wolfram MathWorld
The volumes of any n -dimensional solids can always be simultaneously bisected by a (n-1) -dimensional hyperplane. Proving the theorem for n=2 ...