Newton–Gauss line

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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

Labels used in proof concerning complete quadrilateral

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 Template:Mvar be labeled as in the diagram with diagonals Template:Mvar and their respective midpoints Template:Mvar. Let the midpoints of Template:Mvar be Template:Mvar respectively. Using similar triangles it is seen that Template:Mvar intersects Template:Mvar at Template:Mvar, Template:Mvar intersects Template:Mvar at Template:Mvar and Template:Mvar intersects Template:Mvar at Template:Mvar. Again, similar triangles provide the following proportions,

$\frac{\overline{RL}}{\overline{LQ}}=\frac{\overline{BA}}{\overline{AC}},\frac{\overline{QN}}{\overline{NP}}=\frac{\overline{A^{\prime }{C}^{\prime }}}{\overline{C^{\prime }B}},\frac{\overline{PM}}{\overline{MR}}=\frac{\overline{C{B}^{\prime }}}{\overline{B^{\prime }{A}^{\prime }}}.$

However, the line Template:Math intersects the sides of triangle Template:Math, 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 Template:Mvar are collinear on the sides of triangle Template:Math.

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

Figure 1: An angle equality.

Given any cyclic quadrilateral Template:Mvar, let point Template:Mvar be the point of intersection between the two diagonals Template:Mvar and Template:Mvar. Extend the diagonals Template:Mvar and Template:Mvar until they meet at the point of intersection, Template:Mvar. Let the midpoint of the segment Template:Mvar be Template:Mvar, and let the midpoint of the segment Template:Mvar be Template:Mvar (Figure 1).

Theorem

If the midpoint of the line segment Template:Mvar is Template:Mvar, the Newton–Gauss line of the complete quadrilateral Template:Mvar and the line Template:Mvar determine an angle Template:Math equal to Template:Math.

Proof

First show that the triangles Template:Math are similar.

Since Template:Math and Template:Math, we know Template:Math. Also, $\frac{\overline{BE}}{\overline{PN}}=\frac{\overline{FC}}{\overline{PM}}=2.$

In the cyclic quadrilateral Template:Mvar, these equalities hold:

$\begin{array}{cc}\mathrm{\angle }EDF& =\mathrm{\angle }ADF+\mathrm{\angle }EDA,\\ & =\mathrm{\angle }ACB+\mathrm{\angle }ABC,\\ & =\mathrm{\angle }EAC.\end{array}$

Therefore, Template:Math.

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

$\frac{\overline{BE}}{\overline{FC}}=\frac{2R_1\sin \mathrm{\angle }EDB}{2R_2\sin \mathrm{\angle }FDC}=\frac{R_1}{R_2}=\frac{2R_1\sin \mathrm{\angle }EBD}{2R_2\sin \mathrm{\angle }FCD}=\frac{\overline{DE}}{\overline{DF}}.$

Since Template:Math and Template:Math, this shows the equality $\frac{\overline{PN}}{\overline{PM}}=\frac{\overline{DE}}{\overline{DF}}.$ The similarity of triangles Template:Math follows, and Template:Math.

Remark

If Template:Mvar is the midpoint of the line segment Template:Mvar, it follows by the same reasoning that Template:Math.

Figure 2: Isogonal lines.

Isogonal lines

Theorem

The line through Template:Mvar parallel to the Newton–Gauss line of the complete quadrilateral Template:Mvar and the line Template:Mvar are isogonal lines of Template:Math, that is, each line is a reflection of the other about the angle bisector. (Figure 2)

Proof

Triangles Template:Math are similar by the above argument, so Template:Math. Let Template:Mvar be the point of intersection of Template:Mvar and the line parallel to the Newton–Gauss line Template:Mvar through Template:Mvar.

Since Template:Math and Template:Math Template:Math, and Template:Math.

Therefore,

$\begin{array}{cc}\mathrm{\angle }CE{E}^{\prime }& =\mathrm{\angle }DEF-\mathrm{\angle }{E}^{\prime }EF,\\ & =\mathrm{\angle }PNM-\mathrm{\angle }FNM,\\ & =\mathrm{\angle }PNF=\mathrm{\angle }BEF.\end{array}$

Two cyclic quadrilaterals sharing a Newton-Gauss line

Figure 3: Showing that the quadrilaterals Template:Mvar are cyclic.

Lemma

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

The quadrilaterals Template:Mvar and Template:Mvar are cyclic quadrilaterals.

Proof

Template:Math, as previously shown. The points Template:Mvar and Template:Mvar are the respective circumcenters of the right triangles Template:Math. Thus, Template:Math and Template:Math.

Therefore,

$\begin{array}{cc}\mathrm{\angle }PGN+\mathrm{\angle }PMN& =(\mathrm{\angle }PGF+\mathrm{\angle }FGN)+\mathrm{\angle }PMN\\ & =\mathrm{\angle }PFG+\mathrm{\angle }GFN+\mathrm{\angle }EFD\\ & =180^{\circ }\end{array}.$

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

Figure 4: Showing that the complete quadrilaterals Template:Mvar have the same Newton–Gauss line.

Theorem

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

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

Proof

The two complete quadrilaterals have a shared diagonal, Template:Mvar. Template:Mvar lies on the Newton–Gauss line of both quadrilaterals. Template:Mvar is equidistant from Template:Mvar and Template:Mvar, since it is the circumcenter of the cyclic quadrilateral Template:Mvar.

If triangles Template:Math are congruent, and it will follow that Template:Mvar lies on the perpendicular bisector of the line Template:Mvar. Therefore, the line Template:Mvar contains the midpoint of Template:Mvar, and is the Newton–Gauss line of Template:Mvar.

To show that the triangles Template:Math are congruent, first observe that Template:Mvar is a parallelogram, since the points Template:Mvar are midpoints of Template:Mvar respectively.

Therefore,

$\begin{array}{cc}& \overline{MP}=\overline{QF}=\overline{HQ},\\ & \overline{GP}=\overline{PF}=\overline{MQ},\\ & \mathrm{\angle }MPF=\mathrm{\angle }FQM.\end{array}$

Also note that

$\mathrm{\angle }FPG=2\mathrm{\angle }PBG=2\mathrm{\angle }DBA=2\mathrm{\angle }DCA=2\mathrm{\angle }HCF=\mathrm{\angle }HQF.$

Hence,

$\begin{array}{cc}\mathrm{\angle }MPG& =\mathrm{\angle }MPF+\mathrm{\angle }FPG,\\ & =\mathrm{\angle }FQM+\mathrm{\angle }HQF,\\ & =\mathrm{\angle }HQF+\mathrm{\angle }FQM,\\ & =\mathrm{\angle }HQM.\end{array}$

Therefore, Template:Math and Template:Math are congruent by SAS.

Remark

Due to Template:Math being congruent triangles, their circumcircles Template:Mvar are also congruent.

Relation with the Miquel point

The point at infinity along the Newton-Gauss line is the isogonal conjugate of the Miquel point.

Generalization

Dao Thanh Oai showed a generalization of the Newton-Gauss line.

For a triangle Template:Mvar, let Template:Mvar an arbitrary line and Template:Math the Cevian triangle of an arbitrary point Template:Mvar. Template:Mvar intersects Template:Mvar, and Template:Mvar at Template:Math and Template:Math respectively. Then Template:Math, and Template:Math are colinear.

If Template:Mvar is the centroid of the triangle Template:Mvar, the line is Newton-Gauss line of the quadrilateral composed of Template:Mvar and Template:Mvar.

History

The Newton–Gauss line proof was developed by the two mathematicians it is named after: Sir Isaac Newton and Carl Friedrich Gauss.Template: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.

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

References

• Template:Citation
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External links

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