Nine-point circle

200px|thumb|The nine points

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 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 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), Terquem's circle (after Olry Terquem), 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 of the triangle.

Nine Significant Points of Nine Point Circle

File:Nine-point circle.svg

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 intersection (points ) and the triangle's orthocenter (point ).

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.

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 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 .) Thus, Terquem was the first to use the name nine-point circle.

Tangent circles

right|thumb|250px|The nine-point circle is tangent to the incircle and excircles.

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:<blockquote>... 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...

</blockquote>

The triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point.

Other properties of the nine-point circle

The radius of a triangle's circumcircle is twice the radius of that triangle's nine-point circle.

::<math>\overline{ON} = \overline{NH}.</math>

The nine-point center is one-fourth of the way along the Euler line from the centroid to the orthocenter :

thumb| is a cyclic quadrilateral. is the diagonal triangle of . The point of intersection of the bimedians of belongs to the nine-point circle of .

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). are coaxal.
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 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>

Generalization

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 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 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 is interior to or in a region sharing vertical angles with the triangle, but a nine-point hyperbola occurs when is in one of the three adjacent regions, and the hyperbola is rectangular when P lies on the circumcircle of .

Notes

References

External links

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
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).
Euler circle, Euler circle - the center I, Euler circle - the center II, Euler circle - the radius, Simson line - Euler circle, Simson lines - Euler circle at Interactive Geometry