# Simson line

Last updated The Simson line LN (red) of the triangle ABC with respect to point P on the circumcircle

In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear.  The line through these points is the Simson line of P, named for Robert Simson.  The concept was first published, however, by William Wallace in 1799. 

## Contents

The converse is also true; if the three closest points to P on three lines are collinear, and no two of the lines are parallel, then P lies on the circumcircle of the triangle formed by the three lines. Or in other words, the Simson line of a triangle ABC and a point P is just the pedal triangle of ABC and P that has degenerated into a straight line and this condition constrains the locus of P to trace the circumcircle of triangle ABC.

## Equation

Placing the triangle in the complex plane, let the triangle ABC with unit circumcircle have vertices whose locations have complex coordinates a, b, c, and let P with complex coordinates p be a point on the circumcircle. The Simson line is the set of points z satisfying  :Proposition 4

$2abc{\bar {z}}-2pz+p^{2}+(a+b+c)p-(bc+ca+ab)-{\frac {abc}{p}}=0,$ where an overbar indicates complex conjugation.

## 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 Q are points on the circumcircle, then the angle between the Simson lines of P and Q is half the angle of the arc PQ. In particular, if the points are diametrically opposite, their Simson lines are perpendicular and in this case the intersection of the lines lies on the nine-point circle.
• Letting H denote the orthocenter of the triangle ABC, 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 does not 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 nontrivial 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.
• A quadrilateral that is not a parallelogram has one and only one pedal point, called the Simson point, with respect to which the feet on the quadrilateral are collinear.  The Simson point of a trapezoid is the point of intersection of the two nonparallel sides.  :p. 186
• No convex polygon with at least 5 sides has a Simson line. 

## Proof of existence

The method of proof is to show that $\angle NMP+\angle PML=180^{\circ }$ . $PCAB$ is a cyclic quadrilateral, so $\angle PBA+\angle ACP=\angle PBN+\angle ACP=180^{\circ }$ . $PMNB$ is a cyclic quadrilateral (Thales' theorem), so $\angle PBN+\angle NMP=180^{\circ }$ . Hence $\angle NMP=\angle ACP$ . Now $PLCM$ is cyclic, so $\angle PML=\angle PCL=180^{\circ }-\angle ACP$ . Therefore $\angle NMP+\angle PML=\angle ACP+(180^{\circ }-\angle ACP)=180^{\circ }$ .

# Alternate proof

Whatever the point Z in the adjacent figure is, a + c is 90. Also, whatever the point Z is, c and b are going to be equal. Therefore, we have the following:

a + c = 90

∴ a + b = 90 …(c and b are equal) (1)

Now, consider the measure of angle: a + 90 + b.

If we show that this angle is 180, then the Simpson’s theorem is proved.

From (1) we have, a + 90 + b = 180

Q.E.D.

## Generalizations

### Generalization 1

• Let ABC be a triangle, let a line ℓ go through circumcenter O, and let a point P lie on the circumcircle. Let AP, BP, CP meet ℓ at Ap, Bp, Cp respectively. Let A0, B0, C0 be the projections of Ap, Bp, Cp onto BC, CA, AB, respectively. Then A0, B0, C0 are collinear. Moreover, the new line passes through the midpoint of PH, where H is the orthocenter of ΔABC. If ℓ passes through P, the line coincides with the Simson line.   

### Generalization 2

• Let the vertices of the triangle ABC lie on the conic Γ, and let Q, P be two points in the plane. Let PA, PB, PC intersect the conic at A1, B1, C1 respectively. QA1 intersects BC at A2, QB1 intersects AC at B2, and QC1 intersects AB at C2. Then the four points A2, B2, C2, and P are collinear if only if Q lies on the conic Γ. 

## Related Research Articles A quadrilateral is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners). Other names for quadrilateral include quadrangle, tetragon, and 4-gon. A quadrilateral with vertices , , and is sometimes denoted as . A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted . In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector. In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection. In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:  In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus but it is sometimes attributed to Pythagoras. In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. In geometry, a set of points are said to be concyclic if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, but four or more such points in the plane are not necessarily concyclic. In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.

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