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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