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Newton's Theorem: What is it?
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Newton's and Léon Anne's Theorems
The applet illustrates the following statement (Newton's theorem):
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The center of the circle inscribed into a quadrilateral lies on the line joining the midpoints of the latter's diagonals. |
The line is known as Newton's line. It exists for the quadrilaterals that are not parallelograms.
The statement is an immediate consequence of a theorem of Léon Anne:
Theorem (Pierre-Léon Anne, 1806-1850)
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In any quadrilateral ABCD that is not a parallelogram, the locus of points O such that Area(ΔAOB) + Area(ΔCOD) = Area(ΔBOC) + Area(ΔAOD) is Newton's line of the quadrilateral. |
Proof of Newton's Theorem
For any quadrilateral ABCD circumscribed around a circle
Proof 1 of the theorem of Léon Anne
[F. G.-M, p. 767]. Since E is the midpoint of the diagonal BD, the lengths of the perpendiculars from BD onto the line EF are equal. Therefore, say, triangles BOF and DOF have equal areas. Similarly,
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Area(ΔAOB) = Area(ΔAFB) + Area(ΔAOF) + Area(ΔBOF) and Area(ΔDOC) = Area(ΔDFC) - Area(ΔCOF) - Area(ΔDOF). |
Therefore
| (1) | Area(ΔAOB) + Area(ΔCOD) = Area(ΔAFB) + Area(ΔCFD). |
In the same manner,
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Area(ΔAOD) = Area(ΔAFD) + Area(ΔDOF) - Area(ΔAOF) and Area(ΔBOC) = Area(ΔBFC) - Area(ΔBOF) + Area(ΔCOF). |
Wherefrom
| (2) | Area(ΔAOD) + Area(ΔBOC) = Area(ΔAFD) + Area(ΔCFB). |
But we know that Area(ΔAFB) = Area(ΔAFD) and also
| (3) | Area(ΔAOD) + Area(ΔBOC) = Area(ΔAOB) + Area(ΔCOD). |
Proof 2 of the theorem of Léon Anne
[Honsberger, p. 174-175]
Since a median bisects the area of a triangle, the midpoints of the diagonals of the quadrilateral lie one the locus in question. An observation due to Basil Rannie makes it clear that the locus is a straight line.
The area of a triangle with a fixed base and a moving apex is the linear function of the Cartesian coordinates of the latter. If the apex is shared by two triangles, their total area is still a linear function of its coordinates. But the level curves of a linear function are straight lines.
References
- F. G.-M., Exercices de Géométrie, Jacques Gabay, 1991
- R. Honsberger, More Mathematical Morsels, MAA, 1991
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