Special (notable) straight lines of a quadrilateral
Midlines of a quadrilateral
Let G, I, H, J be the midpoints of the sides of a convex quadrilateral ABCD, and let E, F be the midpoints of its diagonals. Let us call the three segments GH, IJ, EF respectively the first, second, and third midlines of the quadrilateral. The first two of them are also called bimedians .
Points
E, K, F lie on one line, the Newton line
Theorems on the midlines of a quadrilateral
- Generalized Newton's theorem. All three midlines of a quadrilateral intersect at a single point (at the vertex centroid («vertex centroid») of the quadrilateral) and are bisected by it.
- The midpoints E and F of the two diagonals, as well as the vertex centroid K of a convex quadrilateral, lie on one line EF. This line is called the Newton line.
- Note that the Newton — Gauss line coincides with the Newton line, since both pass through the midpoints of the diagonals.
- Varignon's theorem:
- Euler's formula: four times the square of the distance between the midpoints of the diagonals equals the sum of the squares of the sides of the quadrilateral minus the sum of the squares of its diagonals.
- Mathematically, for the figure on the left with the gray quadrilateral ABCD, Euler's formula is written as:
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Newton line
The line obtained by joining the midpoints of the diagonals (
L,
M, and
N) is called the Newton — Gauss line (green)
- If in a quadrilateral the two pairs of opposite sides are not parallel, then the two midpoints of its diagonals lie on a line that passes through the midpoint of the segment joining the two points of intersection of these two pairs of opposite sides (in the figure the points are shown in red). This line is called the Newton line (in the figure it is shown in green). In this case the Newton line is always perpendicular to the Aubert line.
- Points lying on the Newton line, satisfy Anne's theorem.
Orthopolar lines of the orthopoles of vertex triples of a quadrilateral
If a fixed straight line ℓ is given, and any one of the three vertices of the quadrilateral 
is chosen, then all the orthopoles of the given line ℓ with respect to all such triangles lie on one line. This line is called the orthopolar line for the given line ℓ with respect to the quadrilateral .
Special (notable) points of a quadrilateral
Centroid of a quadrilateral
- The four segments, each of which joins a vertex of the quadrilateral to the centroid of the triangle formed by the remaining three vertices, intersect at the centroid of the quadrilateral and are divided by it in the ratio 3:1, counting from the vertices.
- See also the properties of the centroid of a quadrilateral.
Poncelet point of a quadrilateral
Inside a quadrilateral there exists a Poncelet point (see the section "Nine-point circles of triangles inside a quadrilateral").
Miquel point of a quadrilateral
Inside a quadrilateral there exists a Miquel point.
See also
equals the sum of the diagonals of the original quadrilateral.
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and second 
midlines of the quadrilateral by the sine of the angle 
between them, that is
.
.
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