Newton's theorem (quadrilateral)

In Euclidean geometry Newton's theorem states that in every tangential quadrilateral other than a rhombus, the center of the incircle lies on the Newton line.

Let ABCD be a tangential quadrilateral with at most one pair of parallel sides.

Furthermore, let E and F the midpoints of its diagonals AC and BD and P be the center of its incircle.

[1] A tangential quadrilateral with two pairs of parallel sides is a rhombus.

In this case, both midpoints and the center of the incircle coincide, and by definition, no Newton line exists.

According to Anne's theorem, showing that the combined areas of opposite triangles PAD and PBC and the combined areas of triangles PAB and PCD are equal is sufficient to ensure that P lies on EF.