Feuerbach district

Feuerbach circle (M = center point)

The Feuerbach circle or nine-point circle is a special circle in the triangle that is named after Karl Wilhelm Feuerbach . There are nine excellent points on it:

• the midpoints of the sides (D, E, F);
• the base points of the heights (G, H, I);
• the midpoints of the upper height sections (J, L, K) (these are the midpoints of the sections between each triangle corner and the vertical intersection point S of triangle ABC).

special cases

• The Feuerbach circle goes through a corner of the triangle (namely the vertex of the right angle ) when the triangle (Fig. 1) is right-angled .
• The Feuerbach circle touches one side of the triangle (namely the base ) when the triangle (Fig. 2) is isosceles .
• The Feuerbach circle coincides with the inscribed circle if the triangle (Fig. 3) is equilateral .

• 

  Image 1: Right-angled triangle with Feuerbach circle (red), five of nine points visible

• 

  Image 2: Isosceles triangle with Feuerbach circle (red), eight of nine points visible

• 

  Image 3: Equilateral triangle with Feuerbach circle (red) and inscribed circle (green), six of nine points visible

properties

Feuerbach circle, incircle and circles

• The Feuerbach circle touches including the inscribed circle of the triangle and excluding the three circles around the triangle, this property is also known as the Feuerbach theorem. The point at which the Feuerbach circle and the inscribed circle touch is called the Feuerbach point of the triangle. ( Caution: some, mostly German, authors refer to the center of the Feuerbach circle as "Feuerbach point" and accordingly the existence of the Feuerbach circle with the properties described in the introduction as Feuerbach's theorem , see e.g. Schupp)
• The center of the Feuerbach circle lies exactly in the middle between the height intersection and the circumcircle center, i.e. also on Euler's straight line .
• The radius of the Feuerbach circle is half the size of the circumferential radius of the triangle.
  

  Feuerbach district and area
• The Feuerbach circle halves the distance between the height intersection and any point on the circumference.
• If an equilateral (right-angled) hyperbola goes through the corners of a triangle, then its center lies on the Feuerbach circle.
• The center of the Kiepert hyperbola lies on the Feuerbach circle.

Coordinates

Center of the Feuerbach circle ( ) ${\displaystyle X_{5}}$
Trilinear coordinates	${\displaystyle \cos(\beta -\gamma )\,:\,\cos(\gamma -\alpha )\,:\,\cos(\alpha -\beta )}$
Barycentric coordinates	${\displaystyle a\cos(\beta -\gamma )\,:\,b\cos(\gamma -\alpha )\,:\,c\cos(\alpha -\beta )}$

Feuerbachpunkt ( ) ${\displaystyle {X_{1}}_{1}}$
Trilinear coordinates	${\displaystyle 1-\cos(\beta -\gamma )\,:\,1-\cos(\gamma -\alpha )\,:\,1-\cos(\alpha -\beta )}$

history

In Germany, the name Feuerbachkreis has become established instead of the name Neunpunktekreis . The reason for this is the relatively difficult proof from Feuerbach that this circle touches the inscribed circle and the approach circles. In the rest of the world it is mostly called the nine point circle . Historically, the more just term Euler's circle is also common.

Leonhard Euler already showed in 1765 that the six points D to I lie on a circle , that is, he showed that the circle defined by the base points of the heights G, H, I also by the center points of the sides E, F, D ( see Fig. 1–3) (which is why it is sometimes called the Euler Circle). In 1821 Charles Julien Brianchon and Jean Victor Poncelet proved that these six points and three more points lie on the circle, the midpoints of the upper elevation sections J, K, L. Feuerbach proved in 1822 that the originally defined by the base points G, H, I going circle touched the in and out circles and also goes through the side centers E, F, D. He does not mention the other three points J, K, L of the Feuerbach circle. Because of the six points D to I, it is sometimes also called the six-point circle. The Feuerbach circle is also sometimes named after Olry Terquem , who himself coined the term nine-point circle for it in 1842 and gave an analytical proof of Feuerbach's theorem about the inward and outward circles (and rediscovered the additional three points). Another rediscovery of the nine-point circle was made by Jakob Steiner in 1828 and TS Davies in 1827. JS MacKay also found some English authors in his 1892 essay on the history of the nine-point circle who had contributed to the history of the Feuerbach circle before 1821.

See also

• Excellent points in the triangle

literature

• Max Koecher , Aloys Krieg : level geometry . 3. Edition. Springer-Verlag, Berlin 2007, ISBN 978-3-540-49327-3 , pp. 164-167 ( excerpt (Google) )
• Charles S. Ogilvy: Entertaining Geometry ("Excursions in geometry", 1969). 3. Edition. Vieweg Verlag, Braunschweig 1984, ISBN 3-528-28314-9 .
• Hans Schupp: Elementary Geometry . Schöningh, Paderborn 1977, ISBN 3-506-99189-2 (Uni-Taschenbücher 669 Mathematik), pp. 133-135
• John Sturgeon MacKay: History of the Nine Point Circle . Proceedings of the Edinburgh Mathematical Society, Volume 11, 1892, pp. 19-61.

Web links

Commons : Nine-point circle  - collection of images, videos and audio files

• Landesbildungsserver-BW - animated GeoGebra worksheets with slide switches to the Feuerbach district with evidence, conclusions and hints.
• Feuerbachkreis - a visualization of the Feuerbachkreis with GeoGebra
• Feuerbach circle and circles - a visualization of the Feuerbach circle, incircle and circles of the triangle with GeoGebra
• Feuerbachkreis and periphery - a visualization of Feuerbachkreis and periphery with GeoGebra
• Feuerbach circle, height intersection and circumference - a visualization of Feuerbach circle, circumference and a distance from a movable point of the circumference to the height intersection; created with GeoGebra
• Darij Grinberg: Feuerbach circle with a lot of information from the environment (PDF; 190 kB) - a very detailed paper on the Feuerbach circle with many sentences from the environment. A lot of evidence, including a proof of Feuerbach's theorem.
• Eric W. Weisstein : Nine Point Circle . In: MathWorld (English).

Individual evidence

1. ^ Eric Weisstein, Nine point circle, Mathworld
2. ↑ Meyer, Berkhan, Neuere Dreiecksgeometrie, in: Enzyklopädie der Mathematischen Wissenschaften , Volume 3-1-2, 1914, p. 1258
3. ^ Website on the story according to J. MacKay (Proc. Edinburgh Math. Society, Volume 11, 1892, pp. 19–57)
4. ^ Jim Wilson, History of the nine point circle, University of Georgia . He's referring to MacKay
5. ↑ cambridge.org