Simson line

The simsonsche line is an object of the triangle geometry. If the base points of the felled from a point solders on the (possibly extended) sides of a triangle on a common line, then this is referred to as a straight line or simsonsche wallacesche straight and the point when her pole. This is exactly the case when lying on the radius.

The Simson line is erroneously named after the mathematician Robert Simson (1687-1768) named, in whose work but can not find work to Simson line. In reality, she was discovered in 1797 by William Wallace ( 1768-1843 ).

Other properties

Parallel to the Simson- Straight

Each Samson - Just a triangle has three special parallels, each passing through one of the three vertices of the triangle. More precisely, the following theorem holds:

Intersection angle between the Simson line

Looking at a triangle two different points on its perimeter, one obtains two different Simson line. The angle of intersection of these two Simson lines is exactly half as large as the angle formed by the two points with the center of the circumcircle.

Samson - Just when Streckenhalbierende

By connecting the orthocenter of a triangle with a point on the circumcircle of the triangle, so this link is bisected by the associated Simson line.

Family of straight lines

Leaving the Simson- pole wander on the circle, then so has the resulting straight lines of Simson- line one deltoids, also referred to as Steiner- hypocycloid, as an envelope.

Others

If two triangles the same area and their associated Simson- line the same pole, the angle of intersection of the two Simson lines is independent of the choice of the pole. In other words, for all points on the common area of the two triangles results in an equal angle of intersection of the two associated Simson line.

Evidence

Is proved: If on the radius, so the foot points lie on a common straight line. For this purpose, one can show that the following holds.

The foot points and above on the Thales circle. Since circumferential angle ( peripheral angles) are equal to the same arc, it follows

On the other hand, is the premise of a cyclic quadrilateral. Therefore, the opposite angle, and this quadrilateral add up to be. Generally speaking, therefore

The points above on the Thales circle, so that a cyclic quadrilateral is. Similarly as before one closes. Because one obtains

Thus, with

The allegation proved.

Note: The specified evidence refers to the position shown in the sketch position of Höhenfußpunkte. If these are different, the reasons must be varied accordingly.