Newton–Gauss line
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/01/Newton-Gauss_Line_Default_Figure.png/220px-Newton-Gauss_Line_Default_Figure.png)
In geometry, the Newton–Gauss line (or Gauss–Newton line) is the line joining the midpoints of the three diagonals of a complete quadrilateral.
The midpoints of the two diagonals of a convex quadrilateral with at most two parallel sides are distinct and thus determine a line, the Newton line. If the sides of such a quadrilateral are extended to form a complete quadrangle, the diagonals of the quadrilateral remain diagonals of the complete quadrangle and the Newton line of the quadrilateral is the Newton–Gauss line of the complete quadrangle.
Complete quadrilaterals
Any four lines in general position (no two lines are parallel, and no three are concurrent) form a complete quadrilateral. This configuration consists of a total of six points, the intersection points of the four lines, with three points on each line and precisely two lines through each point.[1] These six points can be split into pairs so that the line segments determined by any pair do not intersect any of the given four lines except at the endpoints. These three line segments are called diagonals of the complete quadrilateral.
Existence of the Newton−Gauss line
![Labels used in proof concerning complete quadrilateral](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Labeling_for_a_complete_quadrilateral.svg/350px-Labeling_for_a_complete_quadrilateral.svg.png)
It is a well-known theorem that the three midpoints of the diagonals of a complete quadrilateral are collinear.[2] There are several proofs of the result based on areas [2] or wedge products[3] or, as the following proof, on Menelaus's theorem, due to Hillyer and published in 1920.[4]
Let the complete quadrilateral ABCA'B'C' be labeled as in the diagram with diagonals AA' , BB' and CC' and their respective midpoints, L, M and N. Let the midpoints of BC, CA' and A'B be P, Q and R respectively. Using similar triangles it is seen that QR intersects AA' at L, RP intersects BB' at M and PQ intersects CC' at N. Again, similar triangles provide the following proportions,
However, the line AB'C' intersects the sides of triangle A'BC, so by Menelaus's theorem the product of the terms on the right hand sides is −1. Thus, the product of the terms on the left hand sides is also −1 and again by Menelaus's theorem, the points L, M and N are collinear on the sides of triangle PQR.
Applications to cyclic quadrilaterals
The following are some results that use the Newton–Gauss line of complete quadrilaterals that are associated with cyclic quadrilaterals, based on the work of Barbu and Patrascu.[5]
Equal angles
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/86/Newton-Gauss_Line_Figure_1_subtext.png/220px-Newton-Gauss_Line_Figure_1_subtext.png)
Given any cyclic quadrilateral , let point be the point of intersection between the two diagonals and . Extend the diagonals and until they meet at the point of intersection, . Let the midpoint of the segment be , and let the midpoint of the segment be (Figure 1).
Theorem
If the midpoint of the line segment is , the Newton–Gauss line of the complete quadrilateral and the line determine an angle equal to .
Proof
First show that the triangles and are similar.
Since and , we know . Also,
In the cyclic quadrilateral , these equalities hold:
Therefore,
Let and be the radii of the circumcircles of and , respectively. Apply the law of sines to the triangles, to obtain:
Since and , this shows the equality The similarity of triangles and follows, and
Remark
If is the midpoint of the line segment , it follows by the same reasoning that
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Newton-Gauss_Line_Figure_2_Subtext.png/220px-Newton-Gauss_Line_Figure_2_Subtext.png)
Isogonal lines
Theorem
The line through parallel to the Newton–Gauss line of the complete quadrilateral and the line are isogonal lines of , that is, each line is a reflection of the other about the angle bisector.[5] (Figure 2)
Proof
Triangles and are similar by the above argument, so . Let be the point of intersection of and the line parallel to the Newton–Gauss line through .
Since and , , and .
Therefore,
Two cyclic quadrilaterals sharing a Newton-Gauss line
![](http://upload.wikimedia.org/wikipedia/commons/thumb/9/97/Newton-Gauss_Line_Subtext_3.1.png/220px-Newton-Gauss_Line_Subtext_3.1.png)
Lemma
Let and be the orthogonal projections of the point on the lines and respectively.
The quadrilaterals and are cyclic quadrilaterals.[5]
Proof
, as previously shown. The points and are the respective circumcenters of the right triangles and . Thus, and .
Therefore,
Therefore, is a cyclic quadrilateral, and by the same reasoning, also lies on a circle.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Newton-Gauss_Line_Figure_3_subtext_2.png/220px-Newton-Gauss_Line_Figure_3_subtext_2.png)
Theorem
Extend the lines and to intersect and at and respectively (Figure 4).
The complete quadrilaterals and have the same Newton–Gauss line.[5]
Proof
The two complete quadrilaterals have a shared diagonal, . lies on the Newton–Gauss line of both quadrilaterals. is equidistant from and , since it is the circumcenter of the cyclic quadrilateral .
If triangles and are congruent, and it will follow that lies on the perpendicular bisector of the line . Therefore, the line contains the midpoint of , and is the Newton–Gauss line of .
To show that the triangles and are congruent, first observe that is a parallelogram, since the points and are midpoints of and respectively.
Therefore,
- and
Also note that
Hence,
Therefore, and are congruent by SAS.
Remark
Due to and being congruent triangles, their circumcircles and are also congruent.
History
The Newton–Gauss line proof was developed by the two mathematicians it is named after: Sir Isaac Newton and Carl Friedrich Gauss.[citation needed] The initial framework for this theorem is from the work of Newton, in his previous theorem on the Newton line, in which Newton showed that the center of a conic inscribed in a quadrilateral lies on the Newton–Gauss line.[6]
The theorem of Gauss and Bodenmiller states that the three circles whose diameters are the diagonals of a complete quadrilateral are coaxal.[7]
Notes
- ^ Alperin, Roger C. (6 January 2012). "Gauss–Newton Lines and Eleven Point Conics". Research Gate.
- ^ a b Johnson 2007, p. 62
- ^ Pedoe, Dan (1988) [1970], Geometry A Comprehensive Course, Dover, pp. 46–47, ISBN 0-486-65812-0
- ^ Johnson 2007, p. 152
- ^ a b c d Patrascu, Ion. "Some Properties of the Newton–Gauss Line" (PDF). Forum Geometricorum. Retrieved 29 April 2019.
- ^ Wells, David (1991), The Penguin Dictionary of Curious and Interesting Geometry, Penguin Books, p. 36, ISBN 978-0-14-011813-1
- ^ Johnson 2007, p. 172
References
- Johnson, Roger A. (2007) [1929], Advanced Euclidean Geometry, Dover, ISBN 978-0-486-46237-0
- (available on-line as) Johnson, Roger A. (1929). "Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle". HathiTrust. Retrieved 28 May 2019.
External links
- Bogomonly, Alexander. "Theorem of Complete Quadrilateral: What is it?". Retrieved 11 May 2019.