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 midpoints of
respectively.
Denote the orthocenters of the
respectively.
Denote and
the circles with diameters
and
respectively.
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.
It is clear that is the perpendicular to the line
.
Similarly, one can prove that and
lie on the radical axes of
and
points
and
are collinear.
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