Arbelos

In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the baseline) that contains their diameters. In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the baseline) that contains their diameters. The earliest known reference to this figure is in the Book of Lemmas, where some of its mathematical properties are stated as Propositions 4 through 8.The word arbelos is Greek for 'shoemaker's knife'. Two of the semicircles are necessarily concave, with arbitrary diameters a and b; the third semicircle is convex, with diameter a+b. The area of the arbelos is equal to the area of a circle with diameter H A {displaystyle HA} . Proof: For the proof, reflect the arbelos over the line through the points B {displaystyle B} and C {displaystyle C} , and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters B A {displaystyle BA} A C {displaystyle AC} ) are subtracted from the area of the large circle (with diameter B C {displaystyle BC} ). Since the area of a circle is proportional to the square of the diameter (Euclid's Elements, Book XII, Proposition 2; we do not need to know that the constant of proportionality is π 4 {displaystyle {frac {pi }{4}}} ), the problem reduces to showing that 2 ( A H ) 2 = ( B C ) 2 − ( A C ) 2 − ( B A ) 2 {displaystyle 2(AH)^{2}=(BC)^{2}-(AC)^{2}-(BA)^{2}} . The length ( B C ) {displaystyle (BC)} equals the sum of the lengths ( B A ) {displaystyle (BA)} and ( A C ) {displaystyle (AC)} , so this equation simplifies algebraically to the statement that ( A H ) 2 = ( B A ) ( A C ) {displaystyle (AH)^{2}=(BA)(AC)} . Thus the claim is that the length of the segment A H {displaystyle AH} is the geometric mean of the lengths of the segments B A {displaystyle BA} and A C {displaystyle AC} . Now (see Figure) the triangle B H C {displaystyle BHC} , being inscribed in the semicircle, has a right angle at the point H {displaystyle H} (Euclid, Book III, Proposition 31), and consequently ( H A ) {displaystyle (HA)} is indeed a 'mean proportional' between ( B A ) {displaystyle (BA)} and ( A C ) {displaystyle (AC)} (Euclid, Book VI, Proposition 8, Porism). This proof approximates the ancient Greek argument; Harold P. Boas cites a paper of Roger B. Nelsen who implemented the idea as a proof without words. Let D {displaystyle D} and E {displaystyle E} be the points where the segments B H {displaystyle BH} and C H {displaystyle CH} intersect the semicircles A B {displaystyle AB} and A C {displaystyle AC} , respectively. The quadrilateral A D H E {displaystyle ADHE} is actually a rectangle. The line D E {displaystyle DE} is tangent to semicircle B A {displaystyle BA} at D {displaystyle D} and semicircle A C {displaystyle AC} at E {displaystyle E} . The altitude A H {displaystyle AH} divides the arbelos into two regions, each bounded by a semicircle, a straight line segment, and an arc of the outer semicircle. The circles inscribed in each of these regions, known as the Archimedes' circles of the arbelos, have the same size. The name arbelos comes from Greek ἡ ἄρβηλος he árbēlos or ἄρβυλος árbylos, meaning 'shoemaker's knife', a knife used by cobblers from antiquity to the current day, whose blade is said to resemble the geometric figure.