In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:
- The midpoint of each side of the triangle
- The foot of each altitude
- The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes).<ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref>
The nine-point circle is also known as Feuerbach's circle (after Karl Wilhelm Feuerbach), Euler's circle (after Leonhard Euler), Terquem's circle (after Olry Terquem), the six-points circle, the twelve-points circle, the Template:Mvar-point circle, the medioscribed circle, the mid circle or the circum-midcircle. Its center is the nine-point center of the triangle.<ref>Template:Cite journal Kocik and Solecki (sharers of a 2010 Lester R. Ford Award) give a proof of the Nine-Point Circle Theorem.</ref><ref>Template:Cite book</ref>
Nine Significant Points of Nine Point CircleEdit
The diagram above shows the nine significant points of the nine-point circle. Points Template:Mvar are the midpoints of the three sides of the triangle. Points Template:Mvar are the feet of the altitudes of the triangle. Points Template:Mvar are the midpoints of the line segments between each altitude's vertex intersection (points Template:Mvar) and the triangle's orthocenter (point Template:Mvar).
For an acute triangle, six of the points (the midpoints and altitude feet) lie on the triangle itself; for an obtuse triangle two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point circle.
DiscoveryEdit
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 Template:Mvar.) (At a slightly earlier date, Charles Brianchon and Jean-Victor Poncelet had stated and proven the same theorem.) But soon after Feuerbach, mathematician Olry Terquem 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 Template:Mvar.) Thus, Terquem was the first to use the name nine-point circle.
Tangent circlesEdit
In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. 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 at which the incircle and the nine-point circle touch is called the Feuerbach point.
Other properties of the nine-point circleEdit
- The radius of a triangle's circumcircle is twice the radius of that triangle's nine-point circle.<ref name=PL/>Template:Rp
File:9pcircle03.svg Figure 3
- A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle.
File:9pcircle 04.png Figure 4
- The center Template:Mvar of the nine-point circle bisects a segment from the orthocenter Template:Mvar to the circumcenter Template:Mvar (making the orthocenter a center of dilation to both circles):<ref name=PL>Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.</ref>Template:Rp
- <math>\overline{ON} = \overline{NH}.</math>
- The nine-point center Template:Mvar is one-fourth of the way along the Euler line from the centroid Template:Mvar to the orthocenter Template:Mvar:<ref name=PL/>Template:Rp
- <math>\overline{HN} = 3\overline{NG}.</math>
- Let Template:Math be the nine-point circle of the diagonal triangle of a cyclic quadrilateral. The point of intersection of the bimedians of the cyclic quadrilateral belongs to the nine-point circle.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
- The nine-point circle of a reference triangle is the circumcircle of both the reference triangle's medial triangle (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).<ref name=PL/>Template:Rp
- The center of all rectangular hyperbolas 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 of four points Template:Mvar 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 Template:Mvar represent the common nine-point center and Template:Mvar be an arbitrary point in the plane of the orthocentric system. Then
- <math>\overline{NA}^2 + \overline{NB}^2 + \overline{NC}^2 + \overline{NH}^2 = 3R^2</math>
- where Template:Mvar is the common circumradius; and if
- <math>\overline{PA}^2 + \overline{PB}^2 + \overline{PC}^2 + \overline{PH}^2 = K^2,</math>
- where Template:Mvar is kept constant, then the locus of Template:Mvar is a circle centered at Template:Mvar with a radius <math>\tfrac{1}{2} \sqrt{K^2-3R^2}.</math> As Template:Mvar approaches Template:Mvar the locus of Template:Mvar for the corresponding constant Template:Mvar, collapses onto Template:Mvar the nine-point center. Furthermore the nine-point circle is the locus of Template:Mvar such that
- <math>\overline{PA}^2 + \overline{PB}^2 + \overline{PC}^2 + \overline{PH}^2 = 4R^2.</math>
- 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 Template:Math are given that do not form an orthocentric system, then the nine-point circles of Template:Math concur at a point, the Poncelet point of Template:Mvar. 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, 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 Template:Math are given that form a cyclic quadrilateral, then the nine-point circles of Template:Math 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. 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 Template:Mvar by a factor of –Template:Sfrac and its homothetic center Template:Mvar lies on the line connecting the circumcenter Template:Mvar to the anticenter Template:Mvar where
- <math>\overline{ON} = 2\overline{NM}.</math>
- 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, and the circumcircle of its tangential triangle<ref>Template:Harvtxt</ref> are coaxal.<ref>Template:Harvtxt</ref>
- Trilinear coordinates for the center of the Kiepert hyperbola are
- <math>\frac{(b^2 -c^2)^2}{a} : \frac{(c^2-a^2)^2}{b} : \frac{(a^2-b^2)^2}{c}</math>
- Trilinear coordinates for the center of the Jeřábek hyperbola are
- <math>\cos(A)\sin^2(B-C) : \cos(B)\sin^2(C-A) : \cos(C)\sin^2(A-B)</math>
- Letting Template:Math be a variable point in trilinear coordinates, an equation for the nine-point circle is
- <math>x^2\sin 2A + y^2\sin 2B + z^2\sin 2C-2(yz\sin A + zx\sin B + xy\sin C) = 0.</math>
GeneralizationEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The circle is an instance of a conic section and the nine-point circle is an instance of the general nine-point conic that has been constructed with relation to a triangle Template:Math and a fourth point Template:Mvar, where the particular nine-point circle instance arises when Template:Mvar is the orthocenter of Template:Math. The vertices of the triangle and Template:Mvar determine a complete quadrilateral 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 when Template:Mvar is interior to Template:Math or in a region sharing vertical angles with the triangle, but a nine-point hyperbola occurs when Template:Mvar is in one of the three adjacent regions, and the hyperbola is rectangular when P lies on the circumcircle of Template:Math.
See alsoEdit
- Hart circle, a related construction for circular triangles
- Lester's theorem
- Poncelet point
- Synthetic geometry
NotesEdit
ReferencesEdit
External linksEdit
- Nine-point circle - interactive illustration of the nine-point circle and some of its properties
- "A Javascript demonstration of the nine point circle" at rykap.com
- Encyclopedia of Triangles Centers 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: History of the Nine Point Circle
- Template:Mathworld
- Template:Mathworld
- Nine Point Circle at cut-the-knot
- Interactive Nine Point Circle applet from the Wolfram Demonstrations Project
- Nine-point conic and Euler line generalization at Dynamic Geometry Sketches Generalizes nine-point circle to a nine-point conic with an associated generalization of the Euler line.
- N J Wildberger. Chromogeometry. Discusses the nine-point circle with regard to three different quadratic forms (blue, red, green).