geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, Archimedes' quadruplets are four

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

s associated with an

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 conta ...

. Introduced by Frank Power in the summer of 1998, each have the same

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

Archimedes' twin circles In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three semicircles that have , , and as their diam ...

, making them

Archimedean circle In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. If the arbelos is normed such that the diameter of its outer (largest) half circle has a length of 1 and ' ...

Construction

An arbelos is formed from three collinear points ''A'', ''B'', and ''C'', by the three

semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, radians, or a half-turn). It only has one line of symmetr ...

s with

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

s ''AB'', ''AC'', and ''BC''. Let the two smaller circles have

radii In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is the line segment or ...

''r''₁ and ''r''₂, from which it follows that the larger semicircle has radius ''r'' = ''r''₁+''r''₂. Let the points ''D'' and ''E'' be the center and

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

, respectively, of the semicircle with the radius ''r''₁. Let ''H'' be the midpoint of line ''AC''. Then two of the four quadruplet circles are tangent to line ''HE'' at the point ''E'', and are also tangent to the outer semicircle. The other two quadruplet circles are formed in a symmetric way from the semicircle with radius ''r''₂.

Proof of congruency

According to Proposition 5 of

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

' '' Book of Lemmas'', the common

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of Archimedes' twin circles is: :

\frac.

By the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

: :

\left(HE\right)^2=\left(r_1\right)^2+\left(r_2\right)^2.

Then, create two circles with centers ''J_i''

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

to ''HE'',

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

to the large semicircle at point ''L_i'', tangent to point ''E'', and with equal radii ''x''. Using the

: :

\left(HJ_i\right)^2=\left(HE\right)^2+x^2=\left(r_1\right)^2+\left(r_2\right)^2+x^2

Also: :

HJ_i=HL_i-x=r-x=r_1+r_2-x~

Combining these gives: :

\left(r_1\right)^2+\left(r_2\right)^2+x^2=\left(r_1+r_2-x\right)^2

Expanding, collecting to one side, and factoring: :

2r_1r_2-2x\left(r_1+r_2\right)=0

Solving for ''x'': :

x=\frac=\frac

Proving that each of the Archimedes' quadruplets' areas is equal to each of Archimedes' twin circles' areas.

Construction

Proof of congruency

References

More readings