# Dynamic Geometry 1462: Newton-Line, Newton-Gauss Line, Complete Quadrilateral, Midpoints of
Sides and Diagonals

Given a complete quadrilateral ABDEF (see the figure below) M_{1}, M_{2}, and M_{3} are the midpoints of AC, BD, and EF, respectively. The line segments M_{A}M_{C} and M_{B}M_{D} that connect the midpoints of opposite sides AB with CD and BC with AD
intersect at M_{4}. Prove that M_{1},M_{2},M_{3}, and M_{4} are collinear points.

See solution below

### Newton Line

Newton line is the line that connects the midpoints of the two diagonals in a quadrilateral other than a
parallelogram.

### Newton-Gauss line

The midpoints of the diagonals of a complete quadrilateral lie on a line called Newton-Gauss
line.

### Complete quadrilateral

Complete quadrilateral ABCDEF is the figure determined by four lines, no three of which are concurrent, and their six points of intersection A, B, C, D, E, and F.

### Midpoint

Midpoint of a segment: the point on a line segment dividing it into two segments of equal length.

## Static Diagram of Dynamic Geometry 1462: Newton-Gauss Line

## Poster of the Newton Line and Newton-Gauss Line using iPad Apps

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