Newton–Gauss Line

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. 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

It is a well-known theorem that the three midpoints of the diagonals of a complete quadrilateral are collinear.
There are several proofs of the result based on areas or wedge products or, as the following proof, on Menelaus's theorem, due to Hillyer and published in 1920.

Let the complete quadrilateral be labeled as in the diagram with diagonals and their respective midpoints . Let the midpoints of be respectively. Using similar triangles it is seen that intersects at , intersects at and intersects at . Again, similar triangles provide the following proportions,
:$\frac = \frac, \quad \frac = \frac, \quad \frac = \frac.$
However, the line intersects the sides of triangle , 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 are collinear on the sides of triangle .

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.

Equal angles

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 are similar.

Since and , we know . Also, $\tfrac = \tfrac = 2.$

In the cyclic quadrilateral , these equalities hold:

:$\begin \angle EDF &= \angle ADF + \angle EDA, \\ &= \angle ACB + \angle ABC, \\ &= \angle EAC. \end$

Therefore, .

Let be the radii of the circumcircles of respectively. Apply the law of sines to the triangles, to obtain:

: $\frac =\frac =\frac =\frac = \frac.$

Since and , this shows the equality $\tfrac = \tfrac.$ The similarity of triangles follows, and .

= Remark

=
If is the midpoint of the line segment , it follows by the same reasoning that .

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. (Figure 2)

= Proof

=
Triangles 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,
:$\begin \angle CEE' &= \angle DEF - \angle E'EF, \\ &= \angle PNM - \angle FNM, \\ &= \angle PNF = \angle BEF. \end$

Two cyclic quadrilaterals sharing a Newton-Gauss line

Lemma

Let and be the orthogonal projections of the point on the lines and respectively.

The quadrilaterals and are cyclic quadrilaterals.

= Proof

=
, as previously shown. The points and are the respective circumcenters of the right triangles . Thus, and .

Therefore,

: \begin
\angle PGN + \angle PMN
& = (\angle PGF + \angle FGN) + \angle PMN \\ pt
& = \angle PFG + \angle GFN + \angle EFD \\ pt
&= 180^\circ
\end.

Therefore, is a cyclic quadrilateral, and by the same reasoning, also lies on a circle.

Theorem

Extend the lines to intersect at respectively (Figure 4).

The complete quadrilaterals and have the same Newton–Gauss line.

= 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 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 are congruent, first observe that is a parallelogram, since the points are midpoints of respectively.

Therefore,

:$\begin & \overline = \overline = \overline, \\ & \overline = \overline = \overline, \\ & \angle MPF = \angle FQM. \end$

Also note that

: $\angle FPG = 2 \angle PBG = 2 \angle DBA = 2 \angle DCA = 2 \angle HCF = \angle HQF.$

Hence,

: $\begin \angle MPG &= \angle MPF + \angle FPG, \\ &= \angle FQM + \angle HQF, \\ &= \angle HQF + \angle FQM, \\ &= \angle HQM. \end$

Therefore, and are congruent by SAS.

= Remark

=
Due to being congruent triangles, their circumcircles 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. 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.

The theorem of Gauss and Bodenmiller states that the three circles whose diameters are the diagonals of a complete quadrilateral are coaxal.

Notes

References

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External links

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{{DEFAULTSORT:Newton-Gauss line
Geometry
Quadrilaterals