Spieker point

The Spieker point S of the triangle ABC

As a spike point or spike center of a triangle is referred to incentre the associated middle triangle . You can find the Spieker point by connecting the center points of the sides of the given triangle and bringing the bisector of this center triangle to the intersection. The Spieker point is named after the high school teacher Theodor Spieker (1823-1913).

properties

• The Spieker point of a triangle coincides with the center of gravity of the corresponding triangle circumference, i.e. H. for example, the center of gravity of a wireframe of the triangle.
• The Spieker point lies on a straight line with the inscribed center point , the center of gravity and the nail point . It bisects the connecting distance between the center of the circle and the nail point.
• The Spieker point is the midpoint of the vertical intersection and the Bevan point .
• The Spieker point is the center of a circle that intersects the three circles at right angles.
• The Spieker point lies on the Kiepert hyperbola .

Coordinates

Spieker point (Spieker center, ) ${\displaystyle X_{10}}$
Trilinear coordinates	${\displaystyle bc(b+c)\,:\,ca(c+a)\,:\,ab(a+b)}$
Barycentric coordinates	${\displaystyle (b+c)\,:\,(c+a)\,:\,(a+b)}$

literature

• Hans Walser: Symmetry . MAA, 2000, ISBN 978-0-88385-532-4 , p. 36
• Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 226-227, 249 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry ).

Web links

• Eric W. Weisstein : Spieker Center . In: MathWorld (English).
• The Spiekerpunkt as the focal point of the triangle's circumference
• Medians of a triangle

Individual evidence

1. ↑ Jürgen Flachsmeyer; Rudolf Fritsch; Hans-Christian Reichel (Ed.): Mathematics-Interdisciplinary . (PDF; 177 kB)