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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 points, six lying on the triangle itself (unless the triangle is obtuse). They include:

• The midpoint of each side of the triangle
• The foot of each altitude
• The midpoint of the segment of each altitude from its vertex to the orthocenter (where the three altitudes meet)

The nine-point circle is also known as Feuerbach's circle, Euler's circle, Terquem's circle, the six-points circle, the twelve-points circle, the n-point circle, the medioscribed circle, the mid circle or the circum-midcircle.

Contents 1 Significant points 2 Discovery 3 Tangent circles 4 Other interesting facts 5 See also 6 External links

Figure 1

The diagram above shows the nine significant points of the nine-point circle. Points D, E, and F are the midpoints of the three sides of the triangle. Points G, H, and I are the feet of the altitudes of the triangle. Points J, K, and L are the points on each altitude of the triangle that bisect the line from the altitude's vertex intersection to the triangle's orthocenter (point S).

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 points the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle. (See Fig. 1, points D, E, F, G, H, and I.) (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 points that are the midpoints of the line segments formed between the vertices of the triangle's altitudes and the triangle's orthocenter. (See Fig. 1, points J, K, and L.) Thus, Terquem was the first to use the name nine-point circle.

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. He postulated 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 following image illustrates this theorem.

Figure 2

Thus the point at which the incircle and the nine-point circle touch is often referred to as the Feuerbach point.

• The radius of any nine-point circle is half the length of the radius of the circumcircle of the corresponding triangle.

Figure 3

• A nine-point circle bisects a line going from the corresponding triangle's orthocenter to any point on its circumcircle.

Figure 4

• The center of any nine-point circle (the nine-point center) lies on the corresponding triangle's Euler line, at the midpoint between that triangle's orthocenter and circumcenter.

• If an orthocentric system of four points is given, then the four triangles formed by any combination of three distinct points of that system all have the same nine-point circle.

• The centers of the incircle and excircles of a corresponding 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 of the triangle formed by that orthocentric system are the vertices of the original triangle.

• If arbitrary four points A, B, C, D are given, then the nine-point circle of ABC, BCD, CDA, DAB are concur at a point.

• The center of the Kiepert hyperbola lies on the nine-point circle.

• The center of the Jerabek hyperbola lies on the nine-point circle.

• The Orthopole of line passing through the circumcenter lie on the nine-point circle

• Trilinear coordinates for the nine-point center are cos(B − C) : cos(C − A) : cos(A − B).

• Trilinear coordinates for the Feuerbach point are 1 − cos(B − C) : 1 − cos(C − A) : 1 − cos(A − B).

• Trilinear coordinates for the center of the Kiepert hyperbola are (b² - c²)²/a : (c² - a²)²/b : (a² - b²)²/c

• Trilinear coordinates for the center of the Jerabek hyperbola are cos A sin²(B - C) : cos B sin²(C - A) : cos C sin²(A - B)

• Letting x : y : z be a variable point in trilinear coordinates, an equation for the nine-point circle is

x²sin 2A + y²sin 2B + z²sin 2C - 2(yz sin A + zx sin B + xy sin C) = 0.

• Synthetic geometry

• 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 Jerabek hyperbola as X(125).
• Nine Point Center by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
• History about the nine-point circle based on J.S. MacCay's article from 1892: History of the Nine Point Circle

• Nine Point Circle in Java at cut-the-knot
• Feuerbach's Theorem: a Proof at cut-the-knot
• Special lines and circles in a triangle (requires Java)