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Geometric theorem involving midpoints on a triangle
The midpoint theorem, midsegment theorem, or midline theorem states that if the midpoints of two sides of a triangle are connected, then the resulting
Midpoint_theorem_(triangle)
Topics referred to by the same term
Midpoint theorem may refer to the following mathematical theorems: Midpoint theorem (triangle) Midpoint theorem (conics) Midpoint theorem, describing
Midpoint_theorem
About the midpoint of a chord of a circle, through which two other chords are drawn
Euclidean geometry, the butterfly theorem is a classical result which can be stated as follows: Let M be the midpoint of a chord PQ of a circle, through
Butterfly_theorem
Collinearity of the midpoints of parallel chords in a conic
In geometry, the midpoint theorem describes a property of parallel chords in a conic. It states that the midpoints of parallel chords in a conic are located
Midpoint_theorem_(conics)
Point on a line segment which is equidistant from both endpoints
and passing through its midpoint also passes through the circle's center. The butterfly theorem states that, if M is the midpoint of a chord PQ of a circle
Midpoint
Theorem that any three objects in space can be simultaneously bisected by a plane
mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space
Ham_sandwich_theorem
Theorem in geometry
In Euclidean geometry, Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram, called the Varignon
Varignon's_theorem
On zeros of derivatives of cubic polynomials
that is tangent to each side at the side's midpoint. Marden's theorem still applies: the foci of this midpoint-tangent inellipse are zeroes of the derivative
Marden's_theorem
Theorem concerning ratios of line segments
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry
Intercept_theorem
Geometric line segment whose endpoints lie on a circular arc
= CP · PD (power of a point theorem). The midpoints of a set of parallel chords of a conic are collinear (midpoint theorem for conics). Chords were used
Chord_(geometry)
quarter of the area of the original triangle. This can be proven by the midpoint theorem of triangles and Heron's formula. The orthocenter of the medial triangle
Midpoint_polygon
Relation between the sides of a convex quadrilateral and its diagonals
Euler's quadrilateral theorem or Euler's law on quadrilaterals, named after Leonhard Euler (1707–1783), describes a relation between the sides of a convex
Euler's_quadrilateral_theorem
Relates the length of a median of a triangle to the lengths of its sides
joining a triangle's vertex to the midpoint of the opposite side Ostermann, Alexander; Wanner, Gerhard (2012). "The Theorems of Apollonius–Pappus–Stewart"
Apollonius's_theorem
Theorem on cyclic quadrilateral
intersection of the line EM and the edge AD. Then, the theorem states that F is the midpoint AD. We need to prove that AF = FD. We will prove that both
Brahmagupta_theorem
Theorem about the midpoint of a line connecting squares on two sides of a triangle
Then, the midpoint of the line segment D 1 D 2 {\displaystyle D_{1}D_{2}} is independent of the location of C {\textstyle C} . Van Aubel's theorem Napoleon's
Bottema's_theorem
Theorem in Euclidean geometry
theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. This theorem refers
Mohr–Mascheroni_theorem
Universality of construction using just a straightedge and a single circle with center
In Euclidean geometry, the Poncelet–Steiner theorem is a result about compass and straightedge constructions with certain restrictions. This result states
Poncelet–Steiner_theorem
Exterior angle of a triangle is greater than either of the remote interior angles
exterior angle theorem by: construct the midpoint E of segment AC, draw the ray BE, construct the point F on ray BE so that E is (also) the midpoint of B and
Exterior_angle_theorem
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
In mathematics, Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to
Minkowski's_theorem
Unique ellipse tangent to all 3 midpoints of a given triangle's sides
midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It
Steiner_inellipse
Simple curve of Euclidean geometry
between the midpoint of that chord and the arc of the circle. Given the length y of a chord and the length x of the sagitta, the Pythagorean theorem can be
Circle
Line joining midpoints of a complete quadrilateral's 3 diagonals
diagonals of the complete quadrilateral. It is a well-known theorem that the three midpoints of the diagonals of a complete quadrilateral are collinear
Newton–Gauss_line
On the structure of complete Riemannian manifolds of non-positive sectional curvature
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive
Cartan–Hadamard_theorem
Shape with three sides
inellipse which is interior to the triangle and tangent at the midpoints of the sides. Marden's theorem shows how to find the foci of this ellipse. This ellipse
Triangle
Construction on any polygon that yields a regular polygon with the same number of sides
of van Aubel's theorem. Thus van Aubel's theorem is a special case of the PDN-theorem. In this case the vertices of A1 are the midpoints of the sides of
Petr–Douglas–Neumann_theorem
Branch of mathematics
curves. These two branches are related to each other by the fundamental theorem of calculus. Calculus uses convergence of infinite sequences and infinite
Calculus
Geometric relation between the roots of a polynomial and those of its derivative
zeros, Marden's theorem states that the zeros of P' are the foci of the Steiner inellipse which is the unique ellipse tangent to the midpoints of the triangle
Gauss–Lucas_theorem
Theorem in geometry
opposite edge midpoints are concurrent, and their intersection point is the centroid of the tetrahedron. A specific case of Reusch's theorem where all four
Commandino's_theorem
Existence of a line through two points
The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the
Sylvester–Gallai_theorem
On distance between centers of a triangle
In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by d 2 = R ( R − 2 r ) {\displaystyle
Euler's_theorem_in_geometry
Characterizes spherical triangles with fixed base and area
In spherical geometry, Lexell's theorem holds that every spherical triangle with the same surface area on a fixed base has its apex on a small circle
Lexell's_theorem
High-area shapes can shift to hold many grid points
Blichfeldt's theorem is a mathematical theorem in the geometry of numbers, stating that whenever a bounded set in the Euclidean plane has area A {\displaystyle
Blichfeldt's_theorem
Ranges of numbers contained in each other
one can compare the midpoint of I n {\displaystyle I_{n}} to x {\displaystyle {\sqrt {x}}} in order to determine whether the midpoint is smaller or larger
Nested_intervals
Type of construction
taking advantage of Ceva's theorem and Menelaus's theorem. As a corollary, if there are two points A and B with their midpoint M given, one can construct
Straightedge-only construction
Straightedge-only_construction
Formula used in radio astronomy
The van Cittert–Zernike theorem, named after physicists Pieter Hendrik van Cittert and Frits Zernike, is a formula in coherence theory that states that
Van_Cittert–Zernike_theorem
Approximation technique in integral calculus
numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution. Because
Riemann_sum
Plane curve: conic section
determine the midpoints of two parallel chords, see section on parallel chords. Remark: This property is an affine version of the theorem of two perspective
Parabola
Convex quadrilateral with at least one pair of parallel sides
right triangles, which was used by James Garfield to prove the Pythagorean theorem. A tangential trapezoid is a trapezoid that has an incircle. Four lengths
Trapezoid
On closed convex subsets in Hilbert space
In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector x {\displaystyle x} in a Hilbert
Hilbert_projection_theorem
Plane curve: conic section
H=\left\{P:\left|\left|PF_{2}\right|-\left|PF_{1}\right|\right|=2a\right\}.} The midpoint M {\displaystyle M} of the line segment joining the foci is called the
Hyperbola
Approach to finding numerical solutions of ordinary differential equations
methods, such as the midpoint method also illustrated in the figures, behave more favourably: the global error of the midpoint method is roughly proportional
Euler_method
Triangle containing a 90-degree angle
as its diameter. This is Thales' theorem. The legs and hypotenuse of a right triangle satisfy the Pythagorean theorem: the sum of the areas of the squares
Right_triangle
Mathematical treatise by Euclid
These include the Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many
Euclid's_Elements
Plane curve
^{2}\,\mid \,\left|PF_{2}\right|+\left|PF_{1}\right|=2a\right\}.} The midpoint C {\displaystyle C} of the line segment joining the foci is called the
Ellipse
Perpendicular line segment from a triangle's side to opposite vertex
sides), the altitude having the incongruent side as its base will have the midpoint of that side as its foot. Also the altitude having the incongruent side
Altitude_(triangle)
Geometrical concept relating area and volume
while it is used in some forms, such as its generalization in Fubini's theorem and layer cake representation, results using Cavalieri's principle can
Cavalieri's_principle
Four-sided polygon
where x is the distance between the midpoints of the diagonals. This is sometimes known as Euler's quadrilateral theorem and is a generalization of the parallelogram
Quadrilateral
Theorem about inscribed and circumscribed circles
In geometry, the incenter–excenter lemma is the theorem that the line segment between the incenter and any excenter of a triangle, or between two excenters
Incenter–excenter_lemma
Geometrical theorem
with at most one pair of parallel sides. Furthermore, let E and F the midpoints of its diagonals AC and BD and P be the center of its incircle. Given
Newton's theorem (quadrilateral)
Newton's_theorem_(quadrilateral)
Line segment joining a triangle's vertex to the midpoint of the opposite side
geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly
Median_(geometry)
Point where the incircle and nine-point circle of a triangle are tangent
theorem based on Casey's theorem on the bitangents of four circles tangent to a fifth circle was published by John Casey in 1866; Feuerbach's theorem
Feuerbach_point
Theorem in plane geometry
mapped to points P´, Q´, R´... of another line in the same plane, then the midpoints of the segments PP´, QQ´, RR´... also lie on a line. The proof is easy
Hjelmslev's_theorem
Theorem in group theory
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G {\displaystyle G} has
Stallings theorem about ends of groups
Stallings_theorem_about_ends_of_groups
Theorem in Euclidean geometry
exist since both midpoints of the diagonals coincide with point of intersection of the diagonals. Moreover, the area identity of the theorem holds in this
Anne's_theorem
Non-convex polygons can be made convex by flips
The Erdős–Nagy theorem is a result in discrete geometry stating that a non-convex simple polygon can be made into a convex polygon by a finite sequence
Erdős–Nagy_theorem
Relationship between two lines that meet at a right angle
and the kite. By Brahmagupta's theorem, in an orthodiagonal quadrilateral that is also cyclic, a line through the midpoint of one side and through the intersection
Perpendicular
Method of drawing geometric objects
impossible to take a square root, or mark the midpoint of a segment; but (by the Poncelet–Steiner theorem) given a single circle and its center, they can
Straightedge and compass construction
Straightedge_and_compass_construction
Algorithm for finding a zero of a function
existence of a root in an interval (Descartes' rule of signs, Sturm's theorem, Budan's theorem). They allow extending the bisection method into efficient algorithms
Bisection_method
Division of something into two equal or congruent parts
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
Bisection
Mathematical model of the physical space
intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. One of those is the parallel postulate which relates to parallel
Euclidean_geometry
Line segment from a midpoint of a triangle side which bisects its perimeter
the midpoint of one of the sides of a triangle is parallel to the angle bisectors at the opposite vertex of the triangle. The broken chord theorem of Archimedes
Cleaver_(geometry)
Geometric theorem about isosceles triangles
In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (/ˈpɒnz
Pons_asinorum
Circle constructed from a triangle
These nine points are: The midpoint of each side of the triangle The foot of each altitude The Euler points: the midpoint of the line segment from each
Nine-point_circle
Mathematical theorem
In mathematics, Vincent's theorem—named after Alexandre Joseph Hidulphe Vincent [fr]—is a theorem that isolates the real roots of polynomials with rational
Vincent's_theorem
Distance from an arc's midpoint to the midpoint of its chord
is sagittae) of a circular arc is the distance from the midpoint of the arc to the midpoint of its chord. It is used extensively in architecture when
Sagitta_(geometry)
Quadrilateral whose vertices lie on a circle
Euler's quadrilateral theorem, the circumradius can be expressed in terms of the diagonals p and q, and the distance x between the midpoints of the diagonals
Cyclic_quadrilateral
Point not between two other points
or a proper interval if its endpoints are distinct. The midpoint of an interval is the midpoint of its endpoints. The closed interval [ x , y ] {\displaystyle
Extreme_point
Describes a third square derived from any two squares that share a vertex
triangle. To state the theorem, suppose that ABCD and AB'C'D' are two squares with common vertex A. Let E and G be the midpoints of B'D and D'B respectively
Finsler–Hadwiger_theorem
Quadrilateral with four right angles
isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area. The midpoints of the sides
Rectangle
Non-linear generalization of a Hilbert space
x)^{2}+d(z,y)^{2} \over 2}.} The point m {\displaystyle m} is then the midpoint of x {\displaystyle x} and y : {\displaystyle y:} d ( x , m ) = d ( y
Hadamard_space
Property of points all lying on a single line
trapezoid, the midpoints of the legs are collinear with the incenter. Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if
Collinearity
Quadrilateral with two pairs of parallel sides
circumcircle of ABC, then BGCL is a parallelogram. Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral are the vertices
Parallelogram
Geometric shape
geometric region that further includes all the interior points. By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each of the endpoints
Semicircle
Geometric theorem regarding 3 circles intersecting at a point
therefore is the midpoint of a side of the anticomplementary triangle, and H lies on the perpendicular bisector of this side. Now the midpoints of the sides
Johnson_circles
Triangle with at least two sides congruent
angle bisector from the apex to the base, the median from the apex to the midpoint of the base, the perpendicular bisector of the base within the triangle
Isosceles_triangle
Theorem
Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity). It is one of many conditions for the
Dirichlet–Jordan_test
Property of perpendicular lines through orthocenters
{\displaystyle L_{2}} intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments A 1 A 2 {\displaystyle A_{1}A_{2}} ,
Droz-Farny_line_theorem
Right triangle with a feature making calculations on the triangle easier
an equilateral triangle ABC with side length 2, and with point M as the midpoint of segment BC. Draw an altitude line from A to M. Then ABM is a 30°–60°–90°
Special_right_triangle
Straight line segment that passes through the centre of a circle
straightedge and compass, by finding the midpoint of the segment and then drawing the circle centered at the midpoint through one of the ends of the line segment
Diameter
Probability theory paradox
1/2. The "random midpoint" method: Choose a point anywhere within the circle and construct a chord with the chosen point as its midpoint. The chord is longer
Bertrand paradox (probability)
Bertrand_paradox_(probability)
Plane algebraic curve
Bernoulli is the special case of a Cassini oval which passes through the midpoint between its foci. The lemniscate of Bernoulli results from applying a circle
Lemniscate_of_Bernoulli
Algorithms for zeros of functions
signs (a bracket). Let c = (a + b)/2 be the middle of the interval (the midpoint or the point that bisects the interval). Then either f(a) and f(c), or
Root-finding_algorithm
Type of vector space in math
Theorem 12.6 Reed & Simon 1980, p. 38 Young 1988, p. 23 Clarkson 1936 Rudin 1987, Theorem 4.10 Dunford & Schwartz 1958, II.4.29 Rudin 1987, Theorem 4
Hilbert_space
Fractal curve resembling a blancmange pudding
the blancmange curve is a self-affine fractal curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who
Blancmange_curve
Mathematical treatise by Archimedes
The Method of Mechanical Theorems (Greek: Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as The Method, is one of the major surviving
The Method of Mechanical Theorems
The_Method_of_Mechanical_Theorems
Shape with three equal sides
may be derived from the formula of an isosceles triangle by Pythagoras theorem: the altitude h {\displaystyle h} of a triangle is the square root of the
Equilateral_triangle
Circle that passes through the vertices of a triangle
triangle, the circumcenter always lies at the midpoint of the hypotenuse. This is one form of Thales' theorem. For an obtuse triangle (a triangle with one
Circumcircle
Mathematical relationships
theorem, CF to be the geometric mean from a combination of Thales's theorem (establishing that △ABF is a right triangle) and Geometric mean theorem,
QM–AM–GM–HM_inequalities
Circle tangent to two sides of a triangle and its circumcircle
are the midpoints of Γ {\displaystyle \Gamma } arcs A B {\displaystyle AB} and A C {\displaystyle AC} respectively. The inscribed angle theorem implies
Mixtilinear incircles of a triangle
Mixtilinear_incircles_of_a_triangle
Convex, 4-sided shape with an incircle and a circumcircle
a, b, c, d is bicentric if and only if opposite sides satisfy Pitot's theorem for tangential quadrilaterals and the cyclic quadrilateral property that
Bicentric_quadrilateral
Lines which intersect at a single point
to the midpoint of the opposite side. The three medians meet at the centroid. Perpendicular bisectors are lines running out of the midpoints of each
Concurrent_lines
Line constructed from a triangle
reflection of the foot of the altitude (dropped onto the side line) about the midpoint of the side line being constructed. Furthermore, this point is a tangent
Simson_line
Line intersecting both a vertex and opposite edge of a triangle
Ceva, who proved a theorem about cevians which also bears his name. The length of a cevian can be determined by Stewart's theorem: in the diagram, the
Cevian
Theorem in hyperbolic geometry
they do not intersect and are not limiting parallel. The ultraparallel theorem states that every pair of (distinct) ultraparallel lines has a unique common
Ultraparallel_theorem
First-order method for approximating parallel transport of a vector along a curve
point on γ close to A0, and construct the geodesic X0A1. Let P1 be the midpoint of X0A1 in the sense that the segments X0P1 and P1A1 take an equal affine
Schild's_ladder
Distance-preserving mathematical transformation
following theorem is due to Mazur and Ulam. Definition: The midpoint of two elements x and y in a vector space is the vector 1/2(x + y). Theorem—Let A :
Isometry
Algebraic structure with addition, multiplication, and division
symmetries of field extensions, provides an elegant proof of the Abel–Ruffini theorem that general quintic equations cannot be solved in radicals. Fields serve
Field_(mathematics)
Point found separated from another, given a point pair
line through C is used to find M and N. This fact follows from Desargues theorem. In real projective geometry, harmonic conjugacy can also be defined in
Projective_harmonic_conjugate
Line which touches a circle at exactly one point
Thales' theorem may be used to construct the tangent lines to a point P external to the circle C: A circle is drawn centered on the midpoint M of the
Tangent_lines_to_circles
Equiangular and equilateral polygon
has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon. A regular n-sided polygon
Regular_polygon
MIDPOINT THEOREM
MIDPOINT THEOREM
MIDPOINT THEOREM
MIDPOINT THEOREM
Girl/Female
Indian
New Day; New Morning
Girl/Female
Arabic
Learned; Intelligent
Boy/Male
Arabic, Hindu, Indian, Muslim
Trustworthy; Trusted
Boy/Male
Muslim
King
Boy/Male
Hindu
Boy/Male
Hindu, Indian, Marathi
Endowed with Virtue
Boy/Male
Tamil
Male friend
Girl/Female
British, English, Italian, Latin
Golden Beautiful; Prayer
Girl/Female
Tamil
Sowmya | ஸோவமà¯à®¯à®¾
Peace, Handsome
Boy/Male
Hindu
Wealthy
MIDPOINT THEOREM
MIDPOINT THEOREM
MIDPOINT THEOREM
MIDPOINT THEOREM
MIDPOINT THEOREM
a.
Theorematic.
a.
Alt. of Theorematical
v. t.
To point improperly; to punctuate wrongly.
n.
A theorem or proposition so easy of demonstration as to be almost self-evident.
n.
A numerical coefficient in any particular case of the binomial theorem.
a.
Containing many names or terms; multinominal; as, the polynomial theorem.
n.
The enunciation of a self-evident problem, in distinction from an axiom, which is the enunciation of a self-evident theorem.
n.
One who constructs theorems.
v. t.
To paint ill, or wrongly.
v. t.
To print wrong.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
n.
That which is considered and established as a principle; hence, sometimes, a rule.
n.
A mistake in printing; a deviation from the copy; as, a book full of misprints.
v. t.
To formulate into a theorem.
n.
A statement of a principle to be demonstrated.