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Line constructed from a triangle
the Euler line, named after Leonhard Euler (/ˈɔɪlər/ OY-lər), is a line determined from any triangle that is not equilateral. It is a central line of the
Euler_line
Center of the inscribed circle of a triangle
Greeks, and the only one of the four that does not in general lie on the Euler line. It is the first listed center, X(1), in Clark Kimberling's Encyclopedia
Incenter
mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Triangle containing a 90-degree angle
{\displaystyle 4c^{4}+9a^{2}b^{2}=16m_{a}^{2}m_{b}^{2}.} In a right triangle, the Euler line contains the median on the hypotenuse—that is, it goes through both the
Right_triangle
Straight figure with zero width and depth
have: the Euler line, the Simson lines, and central lines. For a convex quadrilateral with at most two parallel sides, the Newton line is the line that connects
Line_(geometry)
Triangle with at least two sides congruent
which it follows that the Euler line coincides with the axis of symmetry. The incenter of the triangle also lies on the Euler line, something that is not
Isosceles_triangle
Approach to finding numerical solutions of ordinary differential equations
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary
Euler_method
Type of center of a polygon
and has been used to define an Euler line of a quadrilateral. The circumcenter of mass allows us to define an Euler line for simplicial polytopes. Let
Circumcenter_of_mass
Intersection of triangle altitudes
circle all lie on a single line, known as the Euler line. The center of the nine-point circle lies at the midpoint of the Euler line, between the orthocenter
Orthocenter
Inscribed circle of a triangle's medial triangle
throughout the book. The nine-point circle with the Euler line and the Spieker circle with the Nagel line are analogous to each other, but are not duals,
Spieker_circle
Triangle center associated with the nine-point circle
the Euler line of its triangle, at the midpoint between that triangle's orthocenter H and circumcenter O. The centroid G also lies on the same line, 2/3
Nine-point_center
Description of the orientation of a rigid body
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. They
Euler_angles
Property of points all lying on a single line
center of the nine-point circle are collinear, all falling on a line called the Euler line. The de Longchamps point also has other collinearities. Any vertex
Collinearity
Circle constructed from a triangle
The Euler points: the midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments
Nine-point_circle
N.. "The distance from the incenter to the Euler line", Forum Geometricorum 11 (2011): 231–236. L. Euler, "Solutio facilis problematum quorundam geometricorum
List_of_triangle_inequalities
Complex exponential in terms of sine and cosine
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric
Euler's_formula
Polyhedron with four faces
and circumcenter. These points define the Euler line of the tetrahedron that is analogous to the Euler line of a triangle. The nine-point circle of the
Tetrahedron
Mean position of all the points in a shape
circumscribed sphere). These three points define the Euler line of the tetrahedron that is analogous to the Euler line of a triangle. These results generalize to
Centroid
Shape with three sides
(orange), and the circumcenter (green) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint
Triangle
Second-order partial differential equation describing motion of mechanical system
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose
Euler–Lagrange_equation
Geometric property of certain lines with respect to a given triangle
axis of perspectivity of the △ABC and its medial triangle. The Euler line of △ABC is the line passing through the centroid, the circumcenter, the orthocenter
Central_line_(geometry)
Relationship between two lines that meet at a right angle
in triangle geometry. The Euler line of an isosceles triangle is perpendicular to the triangle's base. The Droz-Farny line theorem concerns a property
Perpendicular
4 planar points which are all orthocenters of triangles formed by the other 3
Euler lines of the four possible triangles where the extended line HN is the Euler line of triangle △ABC and the extended line AN is the Euler line of
Orthocentric_system
Orthocenter of a triangle's anticomplementary triangle
notable points on the Euler line", pp. 380–383. Longuet-Higgins, Michael (2000), "A fourfold point of concurrence lying on the Euler line of a triangle", The
De_Longchamps_point
2.71828...; base of natural logarithms
sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant,
E_(mathematical_constant)
Curve whose curvature changes linearly
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the
Euler_spiral
Triangle center: circumcenter of a triangle's excentral triangle
Butterworth. The Bevan point M of triangle △ABC has the same distance from its Euler line e as its incenter I. Their distance is M I ¯ = 2 R 2 − a b c a + b + c
Bevan_point
Conic curves associated with a triangle
parabola is the parabola inscribed in the reference triangle having the Euler line as directrix and the triangle center X110 as focus. The following quote
Kiepert_conics
Circle constructed from a triangle
also contains the triangle's nine-point center and is a subset of the Euler line, which also contains the circumcenter outside the orthocentroidal circle
Orthocentroidal_circle
Topological invariant in mathematics
algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant
Euler_characteristic
Point on a line segment which is equidistant from both endpoints
the orthocenter. These points are all on the Euler line. A midsegment (or midline) of a triangle is a line segment that joins the midpoints of two sides
Midpoint
Triangle centers
Centers, the inner Vecten point is denoted by X(486). The line X485X486 meets the Euler line at the nine-point center of △ABC. The Vecten points lie on
Vecten_points
Difference between logarithm and harmonic series
\ln(x)} or log e ( x ) {\displaystyle \log _{e}(x)} . Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually
Euler's_constant
Number of integers coprime to and less than n
\ln(x)} or log e ( x ) {\displaystyle \log _{e}(x)} . In number theory, Euler's totient function counts the positive integers up to a given integer n {\displaystyle
Euler's_totient_function
Triangles without a right angle
part of the longest side of the triangle. All triangles in which the Euler line is parallel to one side are acute. This property holds for side BC if
Acute_and_obtuse_triangles
Triangle formed by tangents to a given triangle's circumcircle at its vertices
circumcenter of the tangential triangle is on the reference triangle's Euler line, as is the center of similitude of the tangential triangle and the orthic
Tangential_triangle
Trail in a graph that visits each edge once
posthumously in 1873 by Carl Hierholzer. This is known as Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has an even number
Eulerian_path
Soddy line of a triangle is the line that goes through the centers of the two Soddy circles of that triangle. The Soddy line intersects the Euler line in
Soddy_line
Geometric theorem regarding 3 circles intersecting at a point
circumcenter O maps onto the vertices of the Johnson triangle and its Euler line (line passing through O, N, H) generates three lines that are concurrent
Johnson_circles
Point in a triangle that can be seen as its middle under some criteria
three sides. Central line Encyclopedia of Triangle Centers Triangle conic Central triangle Modern triangle geometry Euler line actually the 1st isogonic
Triangle_center
Triangle center minimizing sum of distances to each vertex
lines X(13)X(15) and X(14)X(16) are parallel to the Euler line. The three lines meet at the Euler infinity point, X(30). The points X(13), X(14), the
Fermat_point
Characteristic class of oriented, real vector bundles
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic
Euler_class
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
Point where the incircle and nine-point circle of a triangle are tangent
point and Euler lines", Forum Geometricorum, 6: 191–197, MR 2282236. Vonk, Jan (2009), "The Feuerbach point and reflections of the Euler line", Forum Geometricorum
Feuerbach_point
Geometry of the surface of a sphere
Andalusi scholar Jabir ibn Aflah. Leonhard Euler published a series of important memoirs on spherical geometry: L. Euler, Principes de la trigonométrie sphérique
Spherical_geometry
One of triangle line
orthotransversal of the Feuerbach point is the OI line. The orthotransversal of the Jerabek center is the Euler line. Orthocorrespondents of Fermat points are
Orthotransversal
Triangle geometry term
point as Zeeman–Gossard perspector. Let ABC be any triangle. Let the Euler line of triangle ABC meet the sidelines BC, CA and AB of triangle ABC at D
Gossard_perspector
Four-sided polygon
quasiorthocenter of the convex quadrilateral. These points can be used to define an Euler line of a quadrilateral. In a convex quadrilateral, the quasiorthocenter H
Quadrilateral
Intersection of the 9-point circles of all triangles made from 4 points
ISBN 0-14-011813-6. Vonk, Jan (2009), "The Feuerbach point and reflections of the Euler line" (PDF), Forum Geometricorum, 9: 47–55 Poncelet points and antigonal conjugates
Poncelet_point
Conic plane curve associated with a given triangle
-1<\lambda <{\frac {1}{2}}} , the conics are imaginary. Triangle center Central line Triangle cubic Modern triangle geometry Paris Pamfilos (2021). "Equilaterals
Triangle_conic
{\displaystyle \Delta (t_{a},t_{b},t_{c})=(3/4)\Delta (a,b,c).} The Euler line of an automedian triangle is perpendicular to the median to side a {\displaystyle
Automedian_triangle
Triangle with vertices at midpoints of another triangle's sides
Franzsen, William N. (2011). "The distance from the incenter to the Euler line" (PDF). Forum Geometricorum. 11: 231–236. Chakerian, G. D. "A Distorted
Medial_triangle
Solved problem in mathematics
points O {\displaystyle O} and H {\displaystyle H} are located on the Euler line together with the centroid S {\displaystyle S} the following equation
Sylvester's_triangle_problem
Euler Mathematical Toolbox (or EuMathT; formerly Euler) is a free and open-source numerical software package. It contains a matrix language, a graphical
Euler_Mathematical_Toolbox
Position of something in relation to its surroundings
such as a line, plane or rigid body – is the rotation needed to move the object from a reference placement to its current placement. Euler's rotation theorem
Orientation_(geometry)
Analytic function in mathematics
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable
Riemann_zeta_function
Circle that passes through the vertices of a triangle
collinear with the centroid and orthocenter. The line that passes through all of them is known as the Euler line. The isogonal conjugate of the circumcenter
Circumcircle
Triangle center
triangle ABC lies on the Euler line (the line passing through the centroid, the orthocenter, the de Longchamps point, the Euler centre and the circumcenter)
Exeter_point
Coordinate system that is defined by points instead of vectors
\quad \lambda _{2}=0,\quad \lambda _{3}=0.} The equation of a triangle's Euler line is | λ 1 λ 2 λ 3 1 1 1 tan A tan B tan C | = 0. {\displaystyle
Barycentric_coordinate_system
Intersection of four lines associated with a generalized triangle
Euler line, the Soddy line, the orthic axis and the Gergonne line. Note that the Euler line is orthogonal to the orthic axis and that the Soddy line is
GEOS_circle
Overview of and topical guide to geometry
Isosceles trapezoid Triangle Acute and obtuse triangles Equilateral triangle Euler's line Heron's formula Integer triangle Heronian triangle Isosceles triangle
Outline_of_geometry
Extension of the factorial function
A245886 (Decimal expansion of Gamma(-3/2), where Gamma is Euler's gamma function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane
Gamma_function
Script typeface
AMS Euler is an upright cursive typeface, commissioned by the American Mathematical Society (AMS) and designed and created by Hermann Zapf with the assistance
AMS_Euler
Group of languages
68–76. Euler (2022), pp. 25–26. Seebold (1998), p. 13. Euler (2022), pp. 238, 243. Euler (2022), p. 243. Robinson (1992). Euler (2013), p. 53. Euler (2022)
West_Germanic_languages
Study of angle-preserving transformations
This fact can be used to prove that the Euler line of the intouch triangle of a triangle coincides with its OI line. The proof roughly goes as below: Invert
Inversive_geometry
Scientific educational toy
Euler's Disk, invented between 1987 and 1990 by Joseph Bendik, is a trademarked scientific educational toy. It is used to illustrate and study the dynamic
Euler's_Disk
Mathematical strategy
Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the
Conversion between quaternions and Euler angles
Conversion_between_quaternions_and_Euler_angles
Part of a line that is bounded by two distinct end points; line with two endpoints
a line segment is a part of a straight line that is bounded by two distinct endpoints (its extreme points), and contains every point on the line that
Line_segment
Straight line segment that passes through the centre of a circle
In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It
Diameter
Conjecture on zeros of the zeta function
an Euler product and are not directly related to automorphic representations. At first, the numerical verification that many zeros lie on the line seems
Riemann_hypothesis
Term in motorsports
problem (in the case of an apex in the exact middle of the corner) is the Euler spiral, a curve with a radius that changes at a constant rate. In this solution
Racing_line
Movement with a fixed point is rotation
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the body remains
Euler's_rotation_theorem
Circles tangent to all three sides of a triangle
Franzsen, William N. (2011). "The distance from the incenter to the Euler line" (PDF). Forum Geometricorum. 11: 231–236. MR 2877263. Archived from the
Incircle_and_excircles
Mathematical function
In mathematics, the Euler function is given by ϕ ( q ) = ∏ k = 1 ∞ ( 1 − q k ) , | q | < 1. {\displaystyle \phi (q)=\prod _{k=1}^{\infty }(1-q^{k}),\quad
Euler_function
cusps on the line x = 0, the curves lying on the right hand side of the y-axis. A general expression for particular solutions to the Euler–Tricomi equations
Euler–Tricomi_equation
Procedure for solving ODEs
computational science, Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule), or a similar two-stage
Heun's_method
Geometric model of the physical space
In 1760, Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem.
Three-dimensional_space
Unique ellipse tangent to all 3 midpoints of a given triangle's sides
unique parabola that is tangent to the sides of the triangle and has the Euler line as its directrix. The foci of the Steiner inellipse of a triangle are
Steiner_inellipse
Numeric solution for differential equations
} The explicit midpoint method is sometimes also known as the modified Euler method, the implicit method is the most simple collocation method, and,
Midpoint_method
Lines which intersect at a single point
Longchamps point is the point of concurrence of several lines with the Euler line. Three lines, each formed by drawing an external equilateral triangle
Concurrent_lines
Differential calculus on function spaces
Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such
Calculus_of_variations
Short exact sequence of sheaves on projective space
In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of
Euler_sequence
2012 book by Alfred S. Posamentier and Ingmar Lehmann
geometric objects associated with triangle centers such as the Euler line, Simson line, and nine-point circle. The chapter on areas includes both trigonometric
The_Secrets_of_Triangles
Geometric concept
Longuet-Higgins, Michael S. (2000), "A fourfold point of concurrence lying on the Euler line of a triangle", The Mathematical Intelligencer, 22 (1): 54–59, doi:10
Soddy_circles_of_a_triangle
Unique circle centered at a given triangle's orthocenter
center M) Polar circle d of △ABC (centered at orthocenter H) The centers of these circles relating to △ABC are all collinear–they fall on the Euler line.
Polar_circle_(geometry)
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
Geometric curve associated with a quadrangle
nine-point conic, etc.", Proceedings of the Edinburgh Mathematical Society 24:31–3. Nine-point conic and Euler line generalization at Dynamic Geometry Sketches
Nine-point_conic
Method of integration for rational functions
Euler substitution is a method for evaluating integrals of the form ∫ R ( x , a x 2 + b x + c ) d x , {\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\
Euler_substitution
Method of drawing geometric objects
segment. Drawing a perpendicular line from a point to a line. Bisecting an angle Mirroring a point in a line Constructing a line through a point tangent to
Straightedge and compass construction
Straightedge_and_compass_construction
Plane curve associated with any triangle
pivotal isogonal cubic having its pivot at the intersection of the Euler line with the line at infinity. In Kimberling's Encyclopedia of Triangle Centers,
Neuberg_cubic
About a common point of certain circles defined by an arbitrary triangle
T} of the isogonal conjugate of the point Q {\displaystyle Q} on the Euler line O H {\displaystyle OH} , such that Q H / Q O = 2 t {\displaystyle QH/QO=2t}
Musselman's_theorem
Natural number
(ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than and equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences
34_(number)
Definite integral of a scalar or vector field along a path
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral
Line_integral
Natural number, composite number
the standard form. 40 is an abundant number. Swiss mathematician Leonhard Euler noted 40 prime numbers generated by the quadratic polynomial n 2 + n + 41
40_(number)
Czech mathematician (1845–1931)
hyperbola, the locus of the isogonal conjugate of a point that traverses the Euler line of a triangle. He was honorary member of the Union of Czech mathematicians
Václav_Jeřábek
Transformation of a mathematical sequence
sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the
Binomial_transform
Infinitely detailed mathematical structure
can be conceived of as winding through space differently from an ordinary line – although it is still topologically 1-dimensional, its fractal dimension
Fractal
Notation of differential calculus
named after Joseph Louis Lagrange, although it was in fact invented by Euler and popularized by the former. In Lagrange's notation, a prime mark denotes
Notation_for_differentiation
Divergent series
a meaning" to the series. Other authors have credited Euler with the sum, suggesting that Euler would have extended the relationship between the zeta
1_+_2_+_3_+_4_+_⋯
Natural number
function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08. Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function)"
100
EULER LINE
EULER LINE
Boy/Male
American, Czech, Danish, French, German, Scandinavian, Swedish
Honourable Ruler; Peaceful Ruler; All Ruler; Ever Ruler
Boy/Male
British, English
Wheel Ruler; Circle Ruler
Boy/Male
Muslim
Ruler
Boy/Male
Indian
Ruler
Boy/Male
French, German, Irish
Dominant Ruler; Powerful Ruler
Boy/Male
German, Teutonic
Hardworking Ruler; Home Ruler
Boy/Male
Muslim
Ruler
Boy/Male
American, Anglo, British, Christian, English, German
Wealthy Ruler; Rich Ruler
Boy/Male
American, Chinese, Christian, Danish, French, German, Norse, Scandinavian, Swedish
Ruler; Ruler of the People; Peaceful Ruler; All-ruler; Forever; Alone; Ever Ruler
Boy/Male
Danish, German, Swedish
Island Ruler; Ever Ruler
Boy/Male
German, Swedish
Ever Ruler; Island Ruler
Boy/Male
Christian, German, Norse, Polish, Scandinavian, Swedish
Peaceful Ruler; Forever; Alone; Ruler; All-ruler
Boy/Male
Christian, German, Teutonic
Hard Working Ruler; Industrious Ruler; Home Ruler
Boy/Male
American, British, English
Royal Ruler; King's Ruler
Boy/Male
Indian
Ruler
Boy/Male
German
Powerful Ruler; Army Ruler
Boy/Male
Australian, Dutch, French, German, Italian, Latin, Swiss
Powerful Ruler; Dominant Ruler
Boy/Male
French, German
Wise Ruler; Old Ruler; Long Term Ruler
Boy/Male
Indian
Ruler
Boy/Male
American, Australian, Danish, German
Powerful Ruler; Dominant Ruler
EULER LINE
EULER LINE
Boy/Male
Buddhist, Gujarati, Hindu, Indian, Kannada, Malayalam
Helping Others; Good; Buddhist Angel
Surname or Lastname
English
English : from the Middle English personal name Burret, Old English Burgrǣd, composed of the elements burh, burg ‘fortress’, ‘stronghold’ + rǣd ‘counsel’.English : possibly a nickname for someone with thick and disheveled hair, from Old French b(o)ure ‘coarse woolen cloth’ + Middle English heved ‘head’.
Boy/Male
Latin
Father of Nausicaa.
Girl/Female
Muslim
Loving to her husband woman
Boy/Male
Arabic
Pigeon
Surname or Lastname
English
English : unexplained.
Boy/Male
Hindu, Indian
A Smile
Boy/Male
American, British, Christian, English, Indian, Jamaican
Sculptor; One who Carves Wood; Wood Carver; Carver of Wood or Stone
Female
Norse
 Variant spelling of Old Norse Auðr, AUÃA means "deeply rich."
Boy/Male
Muslim/Islamic
Long-Lived
EULER LINE
EULER LINE
EULER LINE
EULER LINE
EULER LINE
n.
A joint regent or ruler.
a.
Pertaining to Euler, a German mathematician of the 18th century.
n.
A chief or ruler of a deme or district in Greece.
n.
A ruler or governor.
n.
A petty king; a ruler of little power or consequence.
n.
A chief ruler; a potentate. [Obs.] Wyclif.
n.
A ruler; a governor; a prince.
n.
A straight or curved strip of wood, metal, etc., with a smooth edge, used for guiding a pen or pencil in drawing lines. Cf. Rule, n., 7 (a).
n.
A sole or supreme ruler; a sovereign; the highest ruler; an emperor, king, queen, prince, or chief.
a.
The office of ruler; rule; authority; government.
n.
One who rules; one who exercises sway or authority; a governor.
a.
A suffix meaning a ruler, as in monarch (a sole ruler).
a.
One who rules or reigns; a governor; a ruler.
n.
A ruler of one division of a heptarchy.
n.
The mother and ruler of a family or of her descendants; a ruler by maternal right.
n.
One who pules; one who whines or complains; a weak person.
n.
A long, flexble piece of wood sometimes used as a ruler.
n.
A ruler or ruling power.
n.
A Mohammedan title for a ruler; a judge.
n.
A ruler, or sovereign, of a Mohammedan state; specifically, the ruler of the Turks; the Padishah, or Grand Seignior; -- officially so called.