Carter Circle

Carter circle with carter triangle (red),
nail point and height intersection

{\ displaystyle N}

{\ displaystyle H}

{\ displaystyle | AP_ {a} | = BP_ {b} | = | CP_ {c} | = 2r}

The Fuhrmann Circle , named after Wilhelm Fuhrmann (1833–1904), is a special circle on the triangle. For a given triangle with a nail point and a height intersection , the Fuhrmann circle can be defined as the circle that has the line as a diameter. The circle defined in this way is identical to the perimeter of the Fuhrmann triangle belonging to the given triangle . ${\ displaystyle N}$ ${\ displaystyle H}$ ${\ displaystyle NH}$

The radius of the Fuhrmann circle corresponds to the distance between the center points of the inscribed circle and the circumference of the given triangle. With Euler's theorem we get: ${\ displaystyle R_ {F}}$

{\ displaystyle R_ {F} = {\ sqrt {R (R-2r)}}}

Here denotes the radius of the perimeter and the radius of the inscribed circle . ${\ displaystyle R}$ ${\ displaystyle r}$

The carter circle intersects the heights of the triangle next to the common height intersection at another point. Each of these points has the distance from the associated corner point (see drawing). Since the carter circle with these three points together with the nail point, the height intersection and the corner points of the carter triangle has a total of eight special points, it is sometimes also referred to as an eight-point circle . ${\ displaystyle H}$ ${\ displaystyle 2r}$

literature

Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 228-229, 300 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry ).
Ross Honsberger: Episodes in Nineteenth and Twentieth Century Euclidean Geometry . MAA, 1995, pp. 49-52
JA Scott: At Eight-Point Circle . In: The Mathematical Gazette , Volume 86, No. 506 (Jul., 2002), pp. 326–328 ( JSTOR )

Web links

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

Eric W. Weisstein : Fuhrmann circle . In: MathWorld (English).
Carter circle