1u can use vector property i will be much easier than it
We have a quadrilateral ABCD. Let E be the point of intersection of the lines AB and CD. Let F be the point of intersection of the lines AD and BC . Prove the following facts:
a) The midpoints of the segments AC,BD, EF, are collinear.
The line through these midpoints is called the Gauss line of the quadrilateral ABCD.
b) The orthocenters of the triangles ABF ,CDF ,BCE ,ADE are collinear on a line perpendicular to the Gauss line.
The line joining these orthocenters is known as the Aubert line, or, also, as the Steiner line of the quadrilateral ABCD.
c) The circumcircles of the four triangles ABF ,CDF ,BCE ,ADE intersect at a point M.
This point is called the Miquel point of the quadrilateral .
The circumcenters of these four triangles ABF ,CDF ,BCE ,ADElie on a circle which also passes through M .
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