Newton's and Léon Anne's Theorems

Newton Theorem

The center of the circle inscribed into a quadrilateral lies on the line joining the midpoints of the latter's diagonals.

Léon Anne Theorem
In any quadrilateral $ABCD$ that is not a parallelogram, the locus of points $O$ such that

$[AOB]+[COD]=[BOC]+[AOD]$

is Newton's line of the quadrilateral.

Read more »