Arbelos

In geometry, an arbelos is a plane region bounded by three semicircles connected at the corners, 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.

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 ${\displaystyle HA}$.

Proof: For the proof, reflect the line through the points ${\displaystyle B}$ and ${\displaystyle C}$, and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters ${\displaystyle BA}$ ${\displaystyle AC}$) are subtracted from the area of the large circle (with diameter ${\displaystyle BC}$). 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 ${\displaystyle {\frac {\pi }{4}}}$), the problem reduces to showing that ${\displaystyle 2(AH)^{2}=(BC)^{2}-(AC)^{2}-(BA)^{2}}$. The length ${\displaystyle (BC)}$ equals the sum of the lengths ${\displaystyle (BA)}$ and ${\displaystyle (AC)}$, so this equation simplifies algebraically to the statement that ${\displaystyle (AH)^{2}=(BA)(AC)}$. Thus the claim is that the length of the segment ${\displaystyle AH}$ is the geometric mean of the lengths of the segments ${\displaystyle BA}$ and ${\displaystyle AC}$. Now (see Figure) the triangle ${\displaystyle BHC}$, being inscribed in the semicircle, has a right angle at the point ${\displaystyle H}$ (Euclid, Book III, Proposition 31), and consequently ${\displaystyle (HA)}$ is indeed a "mean proportional" between ${\displaystyle (BA)}$ and ${\displaystyle (AC)}$ (Euclid, Book VI, Proposition 8, Porism). This proof approximates the ancient Greek argument; one many find the idea implemented as a proof without words.

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