Newton–Gauss line

These three line segments are called diagonals of the complete quadrilateral.

It is a well-known theorem that the three midpoints of the diagonals of a complete quadrilateral are collinear.

[4] Let the complete quadrilateral ABCA'B'C' be labeled as in the diagram with diagonals AA', BB', CC' and their respective midpoints L, M, N. Let the midpoints of BC, CA', A'B be P, Q, R respectively.

Extend the diagonals AB and CD until they meet at the point of intersection, E. Let the midpoint of the segment EF be N, and let the midpoint of the segment BC be M (Figure 1).

Apply the law of sines to the triangles, to obtain: Since BE = 2 · PN and FC = 2 · PM, this shows the equality

If Q is the midpoint of the line segment FC, it follows by the same reasoning that ∠NMQ = ∠EFA.

Therefore, Let G and H be the orthogonal projections of the point F on the lines AB and CD respectively.

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

Extend the lines GF, HF to intersect EC, EB at I, J respectively (Figure 4).

If triangles △GMP, △HMQ are congruent, and it will follow that M lies on the perpendicular bisector of the line HG.

To show that the triangles △GMP, △HMQ are congruent, first observe that PMQF is a parallelogram, since the points M, P are midpoints of BF, BC respectively.

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

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

Diagonals
Newton-Gauss line through the midpoints L, M, N of the diagonals
Labels used in proof concerning complete quadrilateral
Labels used in proof concerning complete quadrilateral
Figure 1: An angle equality.
Figure 2: Isogonal lines.
Figure 3: Showing that the quadrilaterals MPGN, MQHN are cyclic.
Figure 4: Showing that the complete quadrilaterals EDGHIJ, ABCDEF have the same Newton–Gauss line.