Salinon

The salinon (meaning "salt-cellar" in Greek) is a geometrical figure that consists of four semicircles. It was first introduced by Archimedes in his "Book of Lemmas".citeweb
title="Salinon." From MathWorld--A Wolfram Web Resource
last=Weisstein
first=Eric W.
url=http://mathworld.wolfram.com/Salinon.html
accessdate=2008-04-14]

Construction

Let "O" be the origin on a Cartesian plane. Let "A", "D", "E", and "B" be four points on a line, in that order, with "O" bisecting line "AB". Let "AD" = "EB". Semicircles are drawn above line "AB" with diameters "AB", "AD", and "EB", and another semicircle is drawn below with diameter "DE". A salinon is the figure bounded by these four semicircles. [citation
last=Nelsen
first=Roger B.
chapter=Proof Without Words: The Area of a Salinon
title=Mathematics Magazine
page=130
year=2002
url=http://www.lclark.edu/~mathsci/salinon.pdf]

Properties

Area

Archimedes introduced the salinon in his "Book of Lemmas" by applying Book II, Proposition 10 of Euclid's "Elements". In Archimedes' book, he noted that "the area of the figure bounded by the circumferences of all the semicircles [is] equal to the area of the circle on CF as diameter."cite web
url=http://www.cut-the-knot.org/proofs/Lemma.shtml
title=Salinon: From Archimedes' Book of Lemmas from Interactive Mathematics Miscellany and Puzzles
accessdate=2008-04-15
last=Bogomolny
first=A.
work=from Interactive Mathematics Miscellany and Puzzles]

Namely, the area of the salinon is::$A=frac\{1\}\{4\}pileft(r\_1+r\_2\; ight)^2.$

Proof

Let the radius of the midpoint of "AD" and "EB" be denoted as "G" and "H", respectively. Therefore, "AG" = "GD" = "EH" = "HB" = "r"₁. Because "DO", "OF", and "OE" are all radii to the same semicircle, "DO" = "OF" = "OE" = "r"₂. By segment addition, "AG" + "GD" + "DO" = "OE" + "EH" + "HB" = 2"r"₁ + "r"₂. Since "AB" is the diameter of the salinon, "CF" is the line of symmetry. Because they all are radii of the same semicircle, "AO" = "BO" = "CO" = 2"r"₁ + "r"₂.

Let "P" be the center of the large circle. Because "CO" = 2"r"₁ + "r"₂ and "OF" = "r"₂, "CF" = 2"r"₁ + 2"r"₂. Therefore, the radius of the cirlce is "r"₁ + "r"₂. The area of the circle = π("r"₁ + "r"₂)².

Let "x" = "r"₁ and "y" = "r"₂. The area of the semicircle with diameter "AB" is:

:$AB=frac\{1\}\{2\}pileft(2x+y\; ight)^2$.

The area of the semicircle with diameter "DE" is:

:$DE=frac\{1\}\{2\}pi\; y^2$

The area of each of the semicircles with diameters "AD" and "EB" is

:$AD=EB=frac\{1\}\{2\}pi\; x^2$

Therefore, the area of the salinon is:

:$frac\{1\}\{2\}pileft(left(2x+y\; ight)^2-2x^2+y^2\; ight)=frac\{1\}\{2\}pileft(2x^2+4xy+2y^2\; ight)=pileft(x^2+2xy+y^2\; ight)=pileft(x+y\; ight)^2=pileft(r\_1+r\_2\; ight)^2$

Q.E.D. [citeweb
url=http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Umberger/EMAT6690smu/Essay4smu/Essay4smu.html
last=Umberger
first=Shannon
title=Essay # 4 - The Arbelos and the Salinon
accessdate=2008-04-18]

Arbelos

Should points "D" and "E" converge with "O", it would form an arbelos, another one of Archimedes' creations, with symmetry among the y-axis.

References