The Samson straight line is an object of triangular geometry . If the base points from a point of precipitated solder on the (possibly extended) sides of a triangle on a common straight line, then this straight line as a Simson line or wallacesche straight line and the point as its pole , respectively. This is exactly the case if lies in the perimeter of .
The Samson straight line is erroneously named after the mathematician Robert Simson (1687–1768), in whose work, however, no work on the Samson straight line can be found. In reality, it was discovered in 1797 by William Wallace (1768–1843).
Every Simson straight line of a triangle has three particular parallels, which each run through one of the three corner points of the triangle. More precisely, the following theorem applies:
A triangle , a point P on its circumference and the associated Simson straight line are given. If G AB is the intersection of the perpendicular from P to AB with the circumference, then the straight line CG is parallel to the Simson straight line.
Intersection angle between Simson lines
If you look at two different points on the circumference of a triangle, you get two different Simson lines. The angle of intersection of these two Simson straight lines is exactly half the size of the angle that the two points form with the center of the circumference.
Let and be two points on the perimeter of with a center . Furthermore, let the intersection angle of the two associated Simson lines and . Then applies .
Simson straight line as bisector
If you connect the vertical intersection of a triangle with a point on the circumference of the triangle, this connecting line is halved by the associated Simson straight line.
A triangle , a point on its circumference and the associated Simson straight line are given. H is the orthocenter of , the Simson straight the route cuts in and it is . Also lies on the Feuerbachkreis .
Set of straight lines
Simson lines as tangents of a deltoid
If the Samson pole is allowed to move on the circle, the family of Simson straight lines thus created has a deltoid , also referred to as Steiner's hypocycloid , as an envelope curve .
If two triangles have the same circumcircle and their associated Simson lines have the same pole, the intersection angle of the two Simson lines is independent of the choice of pole. In other words: For all points on the common circumference of the two triangles, the intersection angle of the two associated Simson lines is the same.
Sketch to prove the collinearity of the base points
It is proven: If lies on the perimeter of , the base points lie on a common straight line. One shows that it is true.
Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 137 ff., 206 ff., 243, 251 (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. 43-48, 82-83, 121, 128-136