Gauss line
The Gauss line (or Gauss–Newton line) is the line joining the midpoints of the three diagonals of a complete quadrilateral.
Existence of the Gauss line
The complete quadilateral be given.
Denote
the midpoint
the midpoint
the midpoint
the orthocenter of
the orthocenter of
the orthocenter of
the orthocenter of
the circle with diameter
the circle with diameter
the circle with diameter
the circle with diameter
the circle with diameter
Prove
a) points are collinear;
b)
c) points are collinear.
Proof
Let
is the radical center of
and
lies on the radical axes of
and
is the radical center of
and
lies on the radical axes of
and
Similarly, using circles with diameters and
one can prove that
lies on the radical axes of
and
and on the radical axes of
and
Therefore points
and
are collinear and
is the perpendicular to the line
.
Similarly, one can prove that
and
lie on the radical axes of
and
points H, H_1, H_2$ and H_3 are collinear.