Newton2

Newton line for circumscriptible quadrilaterals

The [Newton] line (see Newton.html ) of a circumscriptible quadrilateral passes through the incenter.

In fact, consider a coordinate system centered at the incenter E of the quadrilateral and the vectors a, b, c, d, pointing to the vertices. The following remarks imply the proof of the theorem:
1) The exterior product (a+b) x (c+d) = 0. In fact this amounts to show that a x c + b x d = - a x d - b x c. This is equivalent to the equality of the areas area(ECD)+area(EAB) = area(EBC)+area(EDA), which, in turn is due to the fact |AD|+|CB| = |AB|+|CD|, which characterizes circumscriptible quadrilaterals.
2) It follows that points P(c+d), E and Q(a+b) are collinear.
3) From 2) follows that the middles K, L of the diagonals and the incenter E are collinear.

See the file BicircularLoci.html for another proof and use of this fact in the case of bicircular quadrangles.

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