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Division of something into two equal or congruent parts
three-dimensional space, bisection is usually done by a bisecting plane, also called the bisector. The perpendicular bisector of a line segment is a line which meets
Bisection
Topics referred to by the same term
Perpendicular bisector construction can refer to: Bisection § Line segment bisector, on the construction of the perpendicular bisector of a line segment
Perpendicular bisector construction
Perpendicular_bisector_construction
In geometry, the perpendicular bisector construction of a quadrilateral is a construction which produces a new quadrilateral from a given quadrilateral
Perpendicular bisector construction of a quadrilateral
Perpendicular_bisector_construction_of_a_quadrilateral
Relationship between two lines that meet at a right angle
Look up perpendicular in Wiktionary, the free dictionary. Definition: perpendicular with interactive animation. How to draw a perpendicular bisector of a
Perpendicular
Universality of construction using just a straightedge and a single circle with center
point M' on the perpendicular bisector of segment BD (constructed by drawing the perpendicular through M), then continuing the construction using point M'
Poncelet–Steiner_theorem
Geometrical theorem relating the lengths of two segments that divide a triangle
then AD is the angle bisector of angle ∠ A. The generalized angle bisector theorem (which is not necessarily an angle bisector theorem, since the angle
Angle_bisector_theorem
Theorem in geometry
diagonals of the trapezoid, such as in the antiparallelogram. Perpendicular bisector construction of a quadrilateral, a different way of forming another quadrilateral
Varignon's_theorem
Four-sided polygon
a pair of opposite edges is removed. Complete quadrangle Perpendicular bisector construction of a quadrilateral Saccheri quadrilateral Types of mesh § Quadrilateral
Quadrilateral
Shape with three sides
section, just a few of the most commonly encountered constructions are explained. A perpendicular bisector of a side of a triangle is a straight line passing
Triangle
construct the line OI" such that OI" is perpendicular to BB' and parallel to B'I". Then, line OA is the angle bisector for ᗉ IAI'. Case 2c: IB' is ultraparallel
Constructions in hyperbolic geometry
Constructions_in_hyperbolic_geometry
Simple curve of Euclidean geometry
perpendicular bisector are: A perpendicular line from the centre of a circle bisects the chord. The line segment through the centre bisecting a chord is
Circle
Point on a line segment which is equidistant from both endpoints
isosceles triangle, the median, altitude, and perpendicular bisector from the base side and the angle bisector of the apex coincide with the Euler line and
Midpoint
Construct all the circles that are tangent to three given circles
The line through P and Q (1) is an angle bisector. Rays have one angle bisector; lines have two, perpendicular to one another. A few basic results are
Special cases of Apollonius' problem
Special_cases_of_Apollonius'_problem
Center of the inscribed circle of a triangle
{\displaystyle {\overline {CF}}} is the bisection of ∠ A C B {\displaystyle \angle {ACB}} . A line that is an angle bisector is equidistant from both of its lines
Incenter
Plane curve: conic section
where the parabola intersects the circle. Another chord BC is the perpendicular bisector of DE and is consequently a diameter of the circle. These two chords
Parabola
Method of drawing geometric objects
Constructing the perpendicular bisector from a segment Finding the midpoint of a segment. Drawing a perpendicular line from a point to a line. Bisecting an angle
Straightedge and compass construction
Straightedge_and_compass_construction
Rules related to the mathematical principles of origami
\mathbf {w} .} A second bisector also exists, perpendicular to the first and passing through pint. Folding along this second bisector will also achieve the
Huzita–Hatori_axioms
Unique ellipse tangent to all 3 midpoints of a given triangle's sides
between the two triangles. From the midpoint m of A'A, draw the perpendicular bisector until it intersects the line BC (or its extension) at point O. Draw
Steiner_inellipse
Conic solid with a polygonal base
centroid), and they are mirror symmetric relative to any perpendicular plane passing through a bisector of the base. Examples are square pyramid and pentagonal
Pyramid_(geometry)
On triangles inscribed in a circle with a diameter as an edge
called the circumcenter, which is the intersection point of the perpendicular bisectors of the triangle. One way of formulating Thales's theorem is: if
Thales's_theorem
Line constructed from a triangle
153–164: Theorem 4. Olga Radko and Emmanuel Tsukerman, "The Perpendicular Bisector Construction, the Isoptic point, and the Simson Line of a Quadrilateral"
Simson_line
Quadrilateral symmetric across a diagonal
two diagonals (the symmetry axis) is the perpendicular bisector of the other, and is also the angle bisector of the two angles it meets. Because of its
Kite_(geometry)
Principle in compass and straightedge constructions
The line DD' is the perpendicular bisector of AB. Thus A is the reflection of B through line DD'. By construction, E is the reflection of C through line
Compass_equivalence_theorem
Problem of constructing equal-area shapes
. He describes the construction of line segment OS as follows. Let AB (Fig.2) be a diameter of a circle whose centre is O. Bisect the arc ACB at C and
Squaring_the_circle
Movement with a fixed point is rotation
that O can be found by intersecting the perpendicular bisector of Aa with the angle bisector of ∠αAa, a construction that might be easier in practice. He
Euler's_rotation_theorem
Construction of an angle equal to one third a given angle
Angle trisection is the construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass
Angle_trisection
2 points about which a triangle can be inverted into an equilateral triangle
axis for each of the three pairs of circles of Apollonius. The perpendicular bisector of line segment S S ′ {\displaystyle SS'} is the Lemoine line, which
Isodynamic_point
Convex polyhedron with regular faces
around the center of a polygon and reflecting an object around the perpendicular bisector of a polygon. The mensuration of polyhedra includes the surface
Johnson_solid
Curve whose normals converge asymptotically
perpendicular bisector are: A perpendicular line from the centre of a horocycle bisects the chord. The line segment through the centre bisecting a chord is
Horocycle
Type of construction
a given point P and perpendicular to any given line l. This construction works regardless of whether P is on line l or not. Bisect any given right angle
Straightedge-only construction
Straightedge-only_construction
Circles in two perpendicular families
intermediate value r = 1, the circle degenerates to a line, the perpendicular bisector of CD. The equation defining these circles as a locus can be generalized
Apollonian_circles
Circles tangent to all three sides of a triangle
or the excenter of A. Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle
Incircle_and_excircles
Circle that passes through the vertices of a triangle
triangle's vertices; its radius is the circumradius. Any point on a perpendicular bisector of one side is equidistant from the two adjacent vertices of the
Circumcircle
Right-angled fractal canopy
include all points of the rectangle; for instance, the points on the perpendicular bisector of the initial line segment (other than the midpoint of this segment)
H_tree
Straight line segment that passes through the centre of a circle
compass, a diameter of a given circle can be constructed as the perpendicular bisector of an arbitrary chord. Drawing two diameters in this way can be
Diameter
Special points within a triangle
circle is at the point where the perpendicular bisector of AB meets the line through point B that is perpendicular to BC). Symmetrically, form a circle
Brocard_points
Point fixed to a body undergoing planar movement
the bisector of B1B2 is y − B y m = tan τ B ( x − B x m ) {\displaystyle y-B_{y}^{m}=\tan \tau _{B}\left(x-B_{x}^{m}\right)} These two bisectors intersect
Instant_centre_of_rotation
Circle tangent to two sides of a triangle and its circumcircle
{\displaystyle {\sqrt {AB\cdot AC}}} and a reflection with respect to the angle bisector on A {\displaystyle A} . Since inversion and reflection are bijective and
Mixtilinear incircles of a triangle
Mixtilinear_incircles_of_a_triangle
Four-dimensional analogue of the cube
a certain length. If another identical line segment is created in a perpendicular direction from itself, it sweeps out and forms a square (2-cube). The
Tesseract
Quadrilateral with sides of equal length
two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral. Its diagonals bisect opposite angles. The first property
Rhombus
Line segment from a midpoint of a triangle side which bisects its perimeter
parallel to the angle bisectors at the opposite vertex of the triangle. The broken chord theorem of Archimedes provides another construction of the cleaver.
Cleaver_(geometry)
Graph of space and time in special relativity
angle bisectors of the axes. The more the relative speed approaches the speed of light the more the axes approach the corresponding angle bisector. The
Spacetime_diagram
On reflection in a spherical mirror
inverse points. Its asymptotic lines are parallel to and perpendicular to the angle bisector of the angle subtended by the given points (or their inverses)
Alhazen's_problem
Theorem in hyperbolic geometry
distance from r and both lie on s. So the perpendicular bisector of D'D (a segment of s) is also perpendicular to r. (If r and s were asymptotically parallel
Ultraparallel_theorem
Line which touches a circle at exactly one point
important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems
Tangent_lines_to_circles
All points whose relative distances to two circles are same
the radical axis is the line segment bisector of M1, M2. In any case the radical axis is a line perpendicular to M 1 M 2 ¯ . {\displaystyle {\overline
Radical_axis
Spheres tangent to a plane inside a cone
intersection of the plane with the cone is symmetric about the perpendicular bisector of the line through F1 and F2 may be counterintuitive, but this
Dandelin_spheres
Model of hyperbolic geometry
part of this line between A and the boundary circle is the bisector. The common perpendicular of two lines is the chord that when extended goes through
Beltrami–Klein_model
Relates the length of a median of a triangle to the lengths of its sides
| , {\displaystyle |AB|=|AC|,} the median A D {\displaystyle AD} is perpendicular to B C {\displaystyle BC} and the theorem reduces to the Pythagorean
Apollonius's_theorem
Cupola with octagonal base
angle. It is also mirror-symmetric relative to any perpendicular plane passing through a bisector of the base. Therefore, it has pyramidal symmetry, the
Square_cupola
Sanskrit grammarian, mathematician and Vedic priest
direction. In modern geometric terms, this construction is equivalent to finding the perpendicular bisector of the east–west segment. Kim Plofker points
Kātyāyana
Cupola with decagonal base
angle. It is also mirror-symmetric relative to any perpendicular plane passing through a bisector of the hexagonal base. Therefore, it has pyramidal symmetry
Pentagonal_cupola
Triangle with integer side lengths
each internal angle bisector of an integer triangle is rational, because the general triangle formula for the internal angle bisector of angle A is 2 b
Integer_triangle
Perpendicular line segment from a triangle's side to opposite vertex
triangle is a line segment through a given vertex (called apex) and perpendicular to a line containing the side or edge opposite the apex. This (finite)
Altitude_(triangle)
Astronomical instrument
zenith and nadir projections, so its center is located on the perpendicular bisection of the segment connecting both points. Indeed, the projection of
Astrolabe
Perpendicular diameters of a circle or hyperbolic-orthogonal diameters of a hyperbola
one diameter is bisected by the other diameter. For example, two diameters of a circle are conjugate if and only if they are perpendicular. For an ellipse
Conjugate_diameters
Polygon with 17 edges
differences to the original: The circle k2 determines the point H instead of the bisector w3. The circle k4 around the point G' (reflection of the point G at m)
Heptadecagon
Figure formed by two rays meeting at a common point
angle bisector with the opposite extended side, are collinear. In a triangle, three intersection points, two between an interior angle bisector and the
Angle
Woodworking angled joint
runs the length of the mating surfaces and another where the spline is perpendicular to the joined edges. Common applications include picture frames, pipes
Miter_joint
Geometric technique for designing mechanical linkages
on the perpendicular bisector of the segment A1A2. Similarly, B is a circling point with a center that is any point on the perpendicular bisector of B1B2
Burmester's_theory
Polyhedron with four faces
circumcenter of a tetrahedron can be found as intersection of three bisector planes. A bisector plane is defined as the plane centered on, and orthogonal to
Tetrahedron
Type of curve in hyperbolic geometry
the line R through M perpendicular to AB must be orthogonal to the axis L. Therefore R is a radius. Also by symmetry, R will bisect the arc AB. The axis
Hypercycle_(geometry)
Convex polygon which can tile the plane by itself
divisors of 360°. Tilings are made by edge-to-edge connections by perpendicular bisectors of the edges of the original uniform lattice, or centroids along
Planigon
Five tiles used in Islamic decorative art
find P to complete the regular pentagon EINPJ. Line DN meets the perpendicular bisector of AB at Q. From Q construct a line parallel to FK to intersect
Girih_tile
Convex quadrilateral with at least one pair of parallel sides
divides the trapezoid into equal areas). The height (or altitude) is the perpendicular distance between the bases. In the case that the two bases have different
Trapezoid
Plane curve: conic section
and B. Next draw the line segment with endpoints A and B and its perpendicular bisector ℓ {\displaystyle \ell } . Construct a hyperbola of eccentricity
Hyperbola
Isometric automorphisms of a hyperbolic space
logarithmic distance is known. For m and n in HP, let b be the perpendicular bisector of the line segment connecting m and n. If b is parallel to the
Hyperbolic_motion
Solid with 2 parallel n-gonal bases connected by n parallelograms
oblique prism is a prism in which the joining edges and faces are not perpendicular to the base faces. Example: a parallelepiped is an oblique prism whose
Prism_(geometry)
Upper-half plane model of hyperbolic non-Euclidean geometry
points. Draw the line segment between the two points. Construct the perpendicular bisector of the line segment. Find its intersection with the x-axis. Draw
Poincaré_half-plane_model
Point in a triangle that can be seen as its middle under some criteria
of all equilateral triangles. The point of concurrence of the perpendicular bisectors of the sides of triangle △ABC is the circumcenter. The trilinear
Triangle_center
Study of angle-preserving transformations
b c {\displaystyle {\sqrt {bc}}} radius, and reflection across A-angle bisector. Its main properties are: Points B and C "swap" places Circle (ABC) becames
Inversive_geometry
Triangle center
an isosceles triangle. An isoscelizer of angle A is a line perpendicular to the bisector of angle A. Isoscelizers were invented by Peter Yff in 1963
Yff_center_of_congruence
Part of a straight line that is bounded by two distinct end points
perpendicular bisectors of the sides (perpendicularly connecting the midpoint of a side to one of the other sides), and the internal angle bisectors (each
Line_segment
Most populous city in Canada
Ontario shoreline, and major north–south arterial roads are roughly perpendicular to the shoreline, though slightly angled north of Eglinton Avenue. This
Toronto
Number, approximately 1.618
made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle
Golden_ratio
Interactive geometry program
addition to the standard compass and ruler tools. These include perpendicular bisector, circle through three points, circumcircular arc through three points
CaRMetal
Intersection of triangle altitudes
new triangle, and the altitudes of the original triangle are the perpendicular bisectors of the new triangle, and therefore concur (at the circumcenter
Orthocenter
Shape with eleven sides
neusis construction and also via two-fold origami. The following construction description is given by T. Drummond from 1800: Draw the radius A B, bisect it
Hendecagon
Relation between sides of a right triangle
pieces do not need to be moved. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse
Pythagorean_theorem
Polygon with 30 edges
with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational
Triacontagon
Shape with same width in all directions
boundary crossed at most twice by any line, and if the line crosses perpendicularly it does so at both crossings, separated by the width. By Barbier's
Curve_of_constant_width
Shape with five sides
periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the vertical axis at point Q. A horizontal line through
Pentagon
Pyramid with a pentagon base
the base. It is also mirror symmetric relative to any perpendicular plane passing through a bisector of the base. It can be represented as the wheel graph
Pentagonal_pyramid
Type of non-Euclidean geometry
horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the line-segment between them. Given any three distinct points
Hyperbolic_geometry
Number, approximately 2.41421
The parallelogram between the pair of grey triangles on the sides has perpendicular diagonals in ratio σ {\displaystyle \sigma } , hence is a silver
Silver_ratio
Civil parish in Azores, Portugal
the regional roadway at Santa Luzia, and running perpendicular towards the regional airport, bisecting the runway in the direction of Santa Rita, until
Lajes_(Praia_da_Vitória)
Plane curve
Because the tangent line is perpendicular to the normal, an equivalent statement is that the tangent is the external angle bisector of the lines to the foci
Ellipse
Triangle center
an isosceles triangle. An isoscelizer of angle A is a line perpendicular to the bisector of angle A. Let △ABC be any triangle. Let P1Q1, P2Q2, P3Q3 be
Congruent_isoscelizers_point
Type of plane partition
equally distant, form a closed half-space, whose boundary is the perpendicular bisector of line segment p j p k {\displaystyle p_{j}p_{k}} . Cell R k {\displaystyle
Voronoi_diagram
Adaptation of the standard Mercator projection
It is tangential to some arbitrarily chosen meridian and its axis is perpendicular to that of the sphere. The x- and y-axes defined on the figure are related
Transverse Mercator projection
Transverse_Mercator_projection
Heritage-listed house in Melbourne, Victoria, Australia
approximately A$4,000. Michael Baker recalled, "I constructed the perpendicular bisector between [the] university and Point Cook, measured off the maximum
Baker_House_(Bacchus_Marsh)
Clinical application of cephalometry (measurement of parts of the head)
cephalometric radiograph is a radiograph of the head taken with the x-ray beam perpendicular to the patient's sagittal plane. Natural head position is a standardized
Cephalometric_analysis
Polygon constructed from another
isosceles triangle is an obtuse isosceles triangle. In the Dorman Luke construction, each face of a dual polyhedron is the dual polygon of the corresponding
Dual_polygon
City in Georgia, United States
Craftsman, and Colonial Revival. The district is bisected by the Georgia/CSX Railroad, which runs perpendicular to the historic commercial core's primary thoroughfare
Lithonia,_Georgia
Polygon shape with eight sides
Draw another diameter GOC, perpendicular to AOE. (Note in passing that A,C,E,G are vertices of a square). Draw the bisectors of the right angles GOA and
Octagon
Curve generated by the projections of a fixed point on the tangents of another curve
the pedal curve of C is the locus of points X so that the line PX is perpendicular to a tangent T to the curve passing through the point X. Conversely
Pedal_curve
Function used in computer graphics
than the general slerp formula is the case when the end vectors are perpendicular, in which case the formula is p0cos θ + p1sin θ. Letting θ = tπ/2, and
Spherical linear interpolation
Spherical_linear_interpolation
Geometry without using coordinates
congruence of triangles. Examples include the Butterfly theorem, Angle bisector theorem, Apollonius' theorem, British flag theorem, Ceva's theorem, Equal
Synthetic_geometry
Planar straight-line mechanisms
in its dimensions, but has the useful property that the motion perpendicularly bisects the fixed base points. It is shaped like a capital A – a stacked
Hart's_inversors
Measuring and marking tool
point at the centre of the circle. Marking lines perpendicular to a curved edge (normal lines). Bisecting square corners to mark a 45° angle. Though some
Combination_square
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PERPENDICULAR BISECTOR-CONSTRUCTION
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PERPENDICULAR BISECTOR-CONSTRUCTION
PERPENDICULAR BISECTOR-CONSTRUCTION
PERPENDICULAR BISECTOR-CONSTRUCTION
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PERPENDICULAR BISECTOR-CONSTRUCTION
PERPENDICULAR BISECTOR-CONSTRUCTION
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