Archimedes' circles
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Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, Archimedes' circles, first created by Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
, are two circles that can be created inside of an arbelos
Arbelos
In geometry, an arbelos is a plane region bounded by a semicircle of diameter 1, connected to semicircles of diameters r and , all oriented the same way and sharing a common baseline. Archimedes is believed to be the first mathematician to study its mathematical properties, as it appears in...
, both having the same area as each other.
Construction
From any three collinear points A, B, and C, one may form an arbelosArbelos
In geometry, an arbelos is a plane region bounded by a semicircle of diameter 1, connected to semicircles of diameters r and , all oriented the same way and sharing a common baseline. Archimedes is believed to be the first mathematician to study its mathematical properties, as it appears in...
, a shape bounded by three semicircle
Semicircle
In mathematics , a semicircle is a two-dimensional geometric shape that forms half of a circle. Being half of a circle's 360°, the arc of a semicircle always measures 180° or a half turn...
s having pairs of these three points as their diameters. All three semicircles must be on the same side of line AC. Archimedes' twin circles are created by drawing a perpendicular line to line AC through the middle point B of the three given points, tangent to the two smaller semicircles. Each of the two circles C1 and C2 is tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
to that line and to the large semicircle; C1 is tangent to one of the smaller semicircles and C2 is tangent to the other smaller semicircle. Each of the two circles is uniquely determined by its three tangencies; constructing each of the twin circles from its tangencies is a special case of the Problem of Apollonius
Problem of Apollonius
In Euclidean plane geometry, Apollonius' problem is to construct circles that are tangent to three given circles in a plane . Apollonius of Perga posed and solved this famous problem in his work ; this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived...
.
Radii of the circles
Because the two circles are congruentCongruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
, they both share the same radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
length. If r = AB/AC, then the radius of either circle is:
Also, according to Proposition 5 of Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
' Book of Lemmas
Book of Lemmas
The Book of Lemmas is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions on circles.-Translations:...
, the common radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of any Archimedean circle is:
where a and b are the radii of two inner semicircles.
Centers of the circles
If r = AB/AC, then the centers to C1 and C2 are:
