Simson's Direct

Simson Straight Triangle ABC

The Simson line is a line passing through the bases of perpendiculars to the sides of a triangle from a point on its circumscribed circle. Its existence is based on the Simson theorem .

Simson's direct bases of perpendiculars lying on the extensions of the sides. The index indicates the side on which the perpendicular is lowered.

Content

1 Simson's theorem
2 History
3 Properties
4 Variations and generalizations

Simson's Theorem

Bases of perpendiculars omitted from an arbitrary point $P$ ${\ displaystyle P}$ circumscribed triangle $A B C$ ${\ displaystyle ABC}$ on its sides or their extensions, lie on one straight line. This line is called the Simson line .

The converse is also true: if the bases of perpendiculars are omitted from the point $P$ ${\ displaystyle P}$ on the sides of the triangle $A B C$ ${\ displaystyle ABC}$ or their continuation, lie on one straight line, then the point $P$ ${\ displaystyle P}$ lies on the circumscribed circle of the triangle.

History

The discovery of this straight line has long been attributed to Robert Simson (1687-1768), but in reality it was discovered only in 1797 by the Scottish mathematician William Wallace . Therefore, along with the traditional name of this line, the historically more just name Wallace's line is often used.

Properties

Let be $H$ ${\ displaystyle H}$ - triangle orthocenter $A B C$ ${\ displaystyle ABC}$ . Then the Simson straight line of an arbitrary point $P$ ${\ displaystyle P}$ divides the line $P H$ ${\ displaystyle PH}$ in half.
On the circumference of a triangle $A B C$ ${\ displaystyle ABC}$ there are exactly three points such that their Simson line is tangent to the Euler circle of the triangle $A B C$ ${\ displaystyle ABC}$ , and these points form a regular triangle . The sides of this triangle are parallel to the sides of the Morley triangle .
Simson's lines of diametrically opposite points of the circumscribed circle are perpendicular.

Variations and generalizations

If from a given point $P$ ${\ displaystyle P}$ circumscribed triangle $A B C$ ${\ displaystyle ABC}$ draw lines at a given oriented angle to the sides, then the three obtained intersection points will lie on one straight line.
Simson's line can be defined for any inscribed $n$ ${\ displaystyle n}$ -gon by induction as follows: Simson straight point $P$ ${\ displaystyle P}$ regarding this $n$ ${\ displaystyle n}$ -gon is called a line containing projections of a point $P$ ${\ displaystyle P}$ on the simson straight $(n-1)$ ${\ displaystyle (n-1)}$ -gons obtained by throwing one vertex $n$ ${\ displaystyle n}$ -gon.
Salmon's theorem
Poderny triangle