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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( is Newton's line of the quadrilateral. |
Proof of Newton's Theorem
For any quadrilateral ABCD circumscribed around a circle 
AOB) + Area(COD) = Area(BOC) + Area(AOD).
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, AOF) = Area(COF).
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Area(AOB) = Area(AFB) + Area(AOF) + Area(BOF) and Area( DOC) = Area(DFC) - Area(COF) - Area(DOF).
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Therefore
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Area(AOB) + Area(COD) = Area(AFB) + Area(CFD).
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In the same manner,
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Area(AOD) = Area(AFD) + Area(DOF) - Area(AOF) and Area( BOC) = Area(BFC) - Area(BOF) + Area(COF).
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Wherefrom
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Area(AOD) + Area(BOC) = Area(AFD) + Area(CFB).
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But we know that Area(AFB) = Area(AFD) and also CFD) = Area(CFB),
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Area(AOD) + Area(BOC) = Area(AOB) + Area(COD).
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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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