![EulerCircle4](https://upload.wikimedia.org/wikipedia/commons/7/7a/EulerCircle4.gif)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the nine-point circle is a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
that can be constructed for any given
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
. It is so named because it passes through nine significant
concyclic points
In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line ar ...
defined from the triangle. These nine
points are:
* The
midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dimen ...
of each side of the triangle
* The
foot
The foot ( : feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is a separate organ at the terminal part of the leg made ...
of each
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
* The midpoint of the
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
from each
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
* Vertex (computer graphics), a data structure that describes the positio ...
of the triangle to the
orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
(where the three altitudes meet; these line segments lie on their respective altitudes).
The nine-point circle is also known as Feuerbach's circle (after
Karl Wilhelm Feuerbach), Euler's circle (after
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
), Terquem's circle (after
Olry Terquem
Olry Terquem (16 June 1782 – 6 May 1862) was a French mathematician. He is known for his works in geometry and for founding two scientific journals, one of which was the first journal about the history of mathematics. He was also the pseudon ...
), the six-points circle, the twelve-points circle, the -point circle, the medioscribed circle, the mid circle or the circum-midcircle. Its center is the
nine-point center
In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle.
It is so called because it is the center of the nine-point circle, a circle t ...
of the triangle.
Nine significant points
The diagram above shows the nine significant points of the nine-point circle. Points are the midpoints of the three sides of the triangle. Points are the feet of the altitudes of the triangle. Points are the midpoints of the line segments between each altitude's
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
* Vertex (computer graphics), a data structure that describes the positio ...
intersection (points ) and the triangle's orthocenter (point ).
For an
acute triangle
An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...
, six of the points (the midpoints and altitude feet) lie on the triangle itself; for an
obtuse triangle
An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...
two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point circle.
Discovery
Although he is credited for its discovery,
Karl Wilhelm Feuerbach did not entirely discover the nine-point circle, but rather the six-point circle, recognizing the significance of the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle. (''See Fig. 1, points'' .) (At a slightly earlier date,
Charles Brianchon
Charles Julien Brianchon (19 December 1783 – 29 April 1864) was a French mathematician and chemist.
Life
He entered into the École Polytechnique in 1804 at the age of eighteen, and studied under Monge, graduating first in his class in 1808 ...
and
Jean-Victor Poncelet
Jean-Victor Poncelet (; 1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work ''Tr ...
had stated and proven the same theorem.) But soon after Feuerbach, mathematician
Olry Terquem
Olry Terquem (16 June 1782 – 6 May 1862) was a French mathematician. He is known for his works in geometry and for founding two scientific journals, one of which was the first journal about the history of mathematics. He was also the pseudon ...
himself proved the existence of the circle. He was the first to recognize the added significance of the three midpoints between the triangle's vertices and the orthocenter. (''See Fig. 1, points'' .) Thus, Terquem was the first to use the name nine-point circle.
Tangent circles
In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to that triangle's three
excircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
s and internally tangent to its
incircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
; this result is known as
Feuerbach's theorem
In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on th ...
. He proved that:
... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle…
The
triangle center
In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...
at which the incircle and the nine-point circle touch is called the
Feuerbach point
In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on th ...
.
Other properties of the nine-point circle
* The radius of a triangle's
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
is twice the radius of that triangle's nine-point circle.
[
''Figure 3''
* A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle.
''Figure 4''
* The center of the nine-point circle bisects a segment from the orthocenter to the ]circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
(making the orthocenter a center of dilation
Dilation (or dilatation) may refer to:
Physiology or medicine
* Cervical dilation, the widening of the cervix in childbirth, miscarriage etc.
* Coronary dilation, or coronary reflex
* Dilation and curettage, the opening of the cervix and surgic ...
to both circles):[Posamentier, Alfred S., and Lehmann, Ingmar. '']The Secrets of Triangles
''The Secrets of Triangles: A Mathematical Journey'' is a popular mathematics book on the geometry of triangles. It was written by Alfred S. Posamentier and , and published in 2012 by Prometheus Books.
Topics
The book consists of ten chapters, ...
'', Prometheus Books, 2012.
::
* The nine-point center is one-fourth of the way along the Euler line
In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, includ ...
from the centroid to the orthocenter :[
::
* Let be the nine-point circle of the diagonal triangle of a ]cyclic quadrilateral
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
. The point of intersection of the bimedians of the cyclic quadrilateral belongs to the nine-point circle.
* The nine-point circle of a reference triangle is the circumcircle of both the reference triangle's medial triangle
In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is n ...
(with vertices at the midpoints of the sides of the reference triangle) and its orthic triangle (with vertices at the feet of the reference triangle's altitudes).[
* The center of all ]rectangular hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
s that pass through the vertices of a triangle lies on its nine-point circle. Examples include the well-known rectangular hyperbolas of Keipert, Jeřábek and Feuerbach. This fact is known as the Feuerbach conic theorem.
* If an orthocentric system
In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and ...
of four points is given, then the four triangles formed by any combination of three distinct points of that system all share the same nine-point circle. This is a consequence of symmetry: the ''sides'' of one triangle adjacent to a vertex that is an orthocenter to another triangle are ''segments'' from that second triangle. A third midpoint lies on their common side. (The same 'midpoints' defining separate nine-point circles, those circles must be concurrent.)
* Consequently, these four triangles have circumcircles with identical radii. Let represent the common nine-point center and be an arbitrary point in the plane of the orthocentric system. Then
::
:where is the common circumradius
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
; and if
::
:where is kept constant, then the locus of is a circle centered at with a radius As approaches the locus of for the corresponding constant , collapses onto the nine-point center. Furthermore the nine-point circle is the locus of such that
::
* The centers of the incircle and excircles of a triangle form an orthocentric system. The nine-point circle created for that orthocentric system is the circumcircle of the original triangle. The feet of the altitudes in the orthocentric system are the vertices of the original triangle.
* If four arbitrary points are given that do not form an orthocentric system, then the nine-point circles of concur at a point, the Poncelet point of . The remaining six intersection points of these nine-point circles each concur with the midpoints of the four triangles. Remarkably, there exists a unique nine-point conic, centered at the centroid of these four arbitrary points, that passes through all seven points of intersection of these nine-point circles. Furthermore, because of the Feuerbach conic theorem mentioned above, there exists a unique rectangular circumconic
In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld ...
, centered at the common intersection point of the four nine-point circles, that passes through the four original arbitrary points as well as the orthocenters of the four triangles.
* If four points are given that form a cyclic quadrilateral
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
, then the nine-point circles of concur at the anticenter of the cyclic quadrilateral. The nine-point circles are all congruent with a radius of half that of the cyclic quadrilateral's circumcircle. The nine-point circles form a set of four Johnson circles
In geometry, a set of Johnson circles comprises three circles of equal radius sharing one common point of intersection . In such a configuration the circles usually have a total of four intersections (points where at least two of them meet): th ...
. Consequently, the four nine-point centers are cyclic and lie on a circle congruent to the four nine-point circles that is centered at the anticenter of the cyclic quadrilateral. Furthermore, the cyclic quadrilateral formed from the four nine-pont centers is homothetic to the reference cyclic quadrilateral by a factor of –½ and its homothetic center lies on the line connecting the circumcenter to the anticenter where
::
* The orthopole of lines passing through the circumcenter lie on the nine-point circle.
* A triangle's circumcircle, its nine-point circle, its polar circle
A polar circle is a geographic term for a conditional circular line (arc) referring either to the Arctic Circle or the Antarctic Circle. These are two of the keynote circles of latitude (parallels). On Earth, the Arctic Circle is currently ...
, and the circumcircle of its tangential triangle
In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertices. Thus the incircle of the ...
are coaxal.
* Trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
for the center of the Kiepert hyperbola *Friedrich Wilhelm August Ludwig Kiepert
Friedrich Wilhelm August Ludwig Kiepert (6 October 1846 – 5 September 1934) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typic ...
are
::
* Trilinear coordinates for the center of the Jeřábek hyperbola are
::
* Letting be a variable point in trilinear coordinates, an equation for the nine-point circle is
::
Generalization
The circle is an instance of a conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
and the nine-point circle is an instance of the general nine-point conic that has been constructed with relation to a triangle and a fourth point , where the particular nine-point circle instance arises when is the orthocenter of . The vertices of the triangle and determine a complete quadrilateral
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six l ...
and three "diagonal points" where opposite sides of the quadrilateral intersect. There are six "sidelines" in the quadrilateral; the nine-point conic intersects the midpoints of these and also includes the diagonal points. The conic is an ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
when is interior to or in a region sharing vertical angles
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles are ...
with the triangle, but a nine-point hyperbola
In Euclidean geometry with triangle , the nine-point hyperbola is an instance of the nine-point conic described by American mathematician Maxime Bôcher in 1892. The celebrated nine-point circle is a separate instance of Bôcher's conic:
:Given ...
occurs when is in one of the three adjacent regions, and the hyperbola is rectangular when P lies on the circumcircle of .
See also
* Lester's theorem
In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle.
The result is named after June Lester, who published it in 1997, and t ...
* Poncelet point
* Synthetic geometry
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compa ...
Notes
References
*
* .
*
*
*
External links
"A Javascript demonstration of the nine point circle"
at rykap.com
by Clark Kimberling. The nine-point center is indexed as X(5), the Feuerbach point, as X(11), the center of the Kiepert hyperbola as X(115), and the center of the Jeřábek hyperbola as X(125).
* History about the nine-point circle based on J.S. MacKay's article from 1892
*
* {{mathworld, urlname=Orthopole, title=Orthopole
Nine Point Circle in Java
at cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Feuerbach's Theorem: a Proof
at cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Special lines and circles in a triangle
by Walter Fendt
Interactive Nine Point Circle applet
from the Wolfram Demonstrations Project
a
Generalizes nine-point circle to a nine-point conic with an associated generalization of the Euler line.
Circles defined for a triangle