Simson line

Simson line

In geometry, given a triangle and a point on its circumcircle, the intersections formed when lines are constructed from the point perpendicular to each of the triangle's sides are collinear. The line through these points is the Simson line, named for Robert Simson. [cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/Extras/Gibson_history_7.html|title=Gibson History 7 - Robert Simson|date= 2008-01-30] The concept was first published, however, by William Wallace. [cite web|url=http://www.cut-the-knot.org/Curriculum/Geometry/Simpson.shtml|title=Simson Line from Interactive Mathematics Miscellany and Puzzles|date= 2008-09-23]

The converse is also true; if the feet of the perpendiculars dropped from a point to the sides of the triangle are collinear, then the point is on the circumcircle. The Simson line of a point is just the pedal triangle of it; the case when that pedal triangle degenerates to a line.

Properties

*The Simson line of a vertex of the triangle is the altitude of the triangle dropped from that vertex, and the Simson line of the point diametrically opposite to the vertex is the side of the triangle opposite to that vertex.

*If P and P' are points on the circumcircle, then the angle between the Simson lines of P and P' is half the angle of the arc PP'. In particular, if the points are diametrically opposite, their Simson lines are perpendicular and in this case the intersection of the lines is on the nine-point circle.

*Let H denote the orthocenter of the triangle ABC, then the Simson line of P bisects the segment PH in a point that lies on the nine-point circle.

*Given two triangles with the same circumcircle, the angle between the Simson lines of a point P on the circumcircle for both triangles doesn't depend of P.

*The set of all Simson lines, when drawn, form an envelope in the shape of a deltoid known as the Steiner deltoid of the reference triangle.

*The construction of the Simson line that coincides with a side of the reference triangle (see first property above) yields a non trivial point on this side line. This point is the reflection of the foot of the altitude (dropped onto the side line) about the midpoint of the side line being constructed. Furthermore this point is a tangent point between the side of the reference triangle and its Steiner deltoid.

Proof of existence

The method of proof is to show that angle MNP + angle PNL = 180^circ. PBCA is a cyclic quadrilateral, so angle PAM + angle CBP = angle PAC + angle CBP = 180^circ. PNMA is a cyclic quadrilateral (Thales' theorem), so angle PAM + angle MNP = 180^circ. Hence angle MNP = angle CBP. Now PLBN is cyclic, so angle PNL = angle PBL = 180^circ - angle CBP. Therefore angle MNP + angle PNL = angle CBP + (180^circ - angle CBP) = 180^circ.

ee also

*Pedal triangle
*Robert Simson

References

External links

* [http://www.cut-the-knot.org/Curriculum/Geometry/Simpson.shtml Simson Line]
* F. M. Jackson and mathworld | urlname = SimsonLine | title = Simson Line


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • simson line — noun Usage: usually capitalized S Etymology: after Robert Simson died 1768 Scottish mathematician : the line joining the feet of the perpendiculars let fall from any point on the circumcircle of a triangle upon the sides of the triangle …   Useful english dictionary

  • Simson (surname) — Simson is a surname, and may refer to* Gabi Simson, American model * Harold Fraser Simson, British composer * Ivan Simson, Brigadier * Martin Eduard von Simson, German jurist and politician * Robert Simson (1687 1768), mathematician and geometer… …   Wikipedia

  • Simson (company) — For other uses, see Simson (disambiguation). Simson Co. (1856) Berlin Suhler Waffen und Fahrzeugwerke (1938) VEB Fahrzeug und Gerätewerk Simson Suhl (1952) VEB Fahrzeug und Jagdwaffenwerk Ernst Thälmann Suhl (1968) Former type manufacturing… …   Wikipedia

  • Robert Simson — (14 October 1687 ndash; 1 October 1768) was a British mathematician and professor of mathematics.The eldest son of John Simson of Kirktonhall, West Kilbride in Ayrshire, Robert Simson was intended for the Church, but the bent of his mind was… …   Wikipedia

  • Droite de Simson — Le point M du cercle circonscrit au triangle (ABC) se projette perpendiculairement sur les trois côtés (prolongés au besoin) en trois points U, V, W alignés. Dans un triangle ABC, soit M un point du plan et U, V et W les projetés orthogonaux de M …   Wikipédia en Français

  • Michelle Simson — MP Member of the Canadian Parliament for Scarborough Southwest In office 2008–2011 Preceded by Tom Wappel Succeeded by …   Wikipedia

  • Pedal triangle — In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle.More specifically, consider a triangle ABC , and a point P that is not one of the vertices A, B, C . Drop perpendiculars from P to the three sides of the …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • William Wallace (mathematician) — William Wallace (September 23, 1768 in Dysart mdash;April 28, 1843 in Edinburgh) was a Scottish mathematician.He mainly worked in the field of geometry and in 1799 became the first to publish the concept of the Simson line, which erroneously was… …   Wikipedia

  • List of triangle topics — This list of triangle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as Pascal s triangle or triangular matrices, or concretely in physical space.… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”