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Simson Line


SimsonLine

The Simson line is the line containing the feet P_1, P_2, and P_3 of the perpendiculars from an arbitrary point P on the circumcircle of a triangle to the sides or their extensions of the triangle. This line was attributed to Simson by Poncelet, but is now frequently known as the Wallace-Simson line since it does not actually appear in any work of Simson (Johnson 1929, p. 137; Coxeter and Greitzer 1967, p. 41; de Guzmán 1999). The inverse statement to that given above, namely that the locus of all points P in the plane of a triangle DeltaABC such that the feet of perpendiculars from the three sides of the triangle is collinear is given by the circumcircle of DeltaABC, is sometimes called the Wallace-Simson theorem (de Guzmán 1999).

The trilinear equation of the Simson line for a point p:q:r lying on the circumcircle, i.e., satisfying

 cpq+bpr+aqr=0,

is

 (p(q+rcosA)(r+qcosA))/(sinA)alpha 
+(q(r+pcosB)(p+rcosB))/(sinB)beta
 +(r(p+qcosC)(q+pcosC))/(sinC)gamma=0

(P. Moses, pers. comm., Jan. 27, 2005).

SimsonLineHP

The Simson line bisects the line HP, where H is the orthocenter (Honsberger 1995, p. 46). Moreover, the midpoint of HP lies on the nine-point circle (Honsberger 1995, pp. 46-47). The Simson lines of two opposite point on the circumcenter of a triangle are perpendicular and meet on the nine-point circle.

The angle between the Simson lines of two points P and P^' is half the angle of the arc PP^'. The Simson line of any polygon vertex is the altitude through that polygon vertex. The Simson line of a point opposite a polygon vertex is the corresponding side. If T_1T_2T_3 is the Simson line of a point T of the circumcircle, then the triangles TT_1T_2 and TA_2A_1 are directly similar.

Simson line deltoid

The envelope of the Simson lines of a triangle is a deltoid (Butchart 1939; Wells 1991, pp. 155 and 230). The area of the deltoid is half the area of the circumcircle (Wells 1991, p. 230), and the first Morley triangle of the starting triangle has the same orientation as the deltoid. Each side of the triangle is tangent to the deltoid at a point whose distance from the midpoint of the side equals the chord of the nine-point circle cut off by that side (Wells 1991, p. 231). If a line L is the Simson line of a point P on the circumcircle of a triangle, then P is called the Simson line pole of L (Honsberger 1995, p. 128).

The altitudes of a reference triangle are Simson lines whose Simson line poles are the vertices of the reference triangle. Furthermore, the sides of the reference triangle are also Simson lines whose Simson line poles are the reflections of the vertices of the reference triangle about its circumcenter. Note also that the nontrivial perpendicular feet from these reflective vertices intersect the sides of the reference triangle at points that are the tangents to the Steiner deltoid.


See also

Circumcircle, Rigby Points, Simson Line Pole, Steiner Deltoid

Portions of this entry contributed by Frank Jackson

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References

Baker, H. F. An Introduction to Plane Geometry. London: Cambridge University Press, 1963.Butchart, J. H. "The Deltoid Regarded as the Envelope of Simson Lines." Amer. Math. Monthly 46, 85-86, 1939.Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 164, 1888.Chou, S.-C. "Proving Elementary Geometry Theorems Using Wu's Algorithm." Contemporary Math. 29, 243-286, 1984.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 49, 1971.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Coxeter, H. S. M. and Greitzer, S. L. "Simson Lines" and "More on Simson Lines." §2.5 and 2.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 40-41 and 43-45, 1967.de Guzmán, M. "An Extension of the Wallace-Simson Theorem: Projecting in Arbitrary Directions." Amer. Math. Monthly 106, 574-580, 1999.Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 46-48, 1928.F. Gabriel-Marie. Exercices de Géométrie. Tours, France: Maison Mame, p. 329, 1912.Gallatly, W. "The Simson Line." Ch. 4 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 24-36, 1913.Honsberger, R. "The Simson Line" and "Simson Lines." §5.2 and 8.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 43-44 and 82-83, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 137-139, 1929.Patterson, B. C. "The Triangle: Its Deltoids and Foliates." Amer. Math. Monthly 47, 11-18, 1940.Ramler, O. J. "The Orthopole Loci of Some One-Parameter Systems of Lines Referred to a Fixed Triangle." Amer. Math. Monthly 37, 130-136, 1930.van Horn, C. E. "The Simson Quartic of a Triangle." Amer. Math. Monthly 45, 434-437, 1938.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 155 and 230-231, 1991.

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Simson Line

Cite this as:

Jackson, Frank and Weisstein, Eric W. "Simson Line." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SimsonLine.html

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