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An arbelos (grey region) Arbelos sculpture in Kaatsheuvel, Netherlands 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 twin circles (red) of an arbelos (grey) In geometry, the twin circles are two special circles associated with an arbelos.An arbelos is determined by three collinear points A, B, and C, and is the curvilinear triangular region between the three semicircles that have AB, BC, and AC as their diameters.
An arbelos is formed from three collinear points A, B, and C, by the three semicircles with diameters AB, AC, and BC. Let the two smaller circles have radii 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, respectively, of the semicircle with the radius r 1.
For a semicircle with a diameter of a + b, the length of its radius is the arithmetic mean of a and b (since the radius is half of the diameter). The geometric mean can be found by dividing the diameter into two segments of lengths a and b, and then connecting their common endpoint to the semicircle with a segment perpendicular to the diameter ...
If and are the radii of the small semicircles of the arbelos, the radius of an Archimedean circle is equal to = + This radius is thus = +.. The Archimedean circle with center (as in the figure at right) is tangent to the tangents from the centers of the small semicircles to the other small semicircle.
An arbelos is a shape bounded by three mutually-tangent semicircular arcs with collinear endpoints, with the two smaller arcs nested inside the larger one; let the endpoints of these three arcs be (in order along the line containing them) A, B, and C.
As noted above, the inversion centered at point A transforms the arbelos circles C U, C V into two parallel lines, and the circles of the Pappus chain into a stack of equally sized circles sandwiched between the two parallel lines. Hence, the points of tangency between the transformed circles lie on a line midway between the two parallel lines.
The Bankoff circle is formed from three semicircles that create an arbelos. A circle C 1 is then formed tangent to each of the three semicircles, as an instance of the problem of Apollonius . Another circle C 2 is then created, through three points: the two points of tangency of C 1 with the smaller two semicircles, and the point where the two ...