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. [1]
The earliest known reference to this figure is in Archimedes's Book of Lemmas , where some of its mathematical properties are stated as Propositions 4 through 8. [2] The word arbelos is Greek for 'shoemaker's knife'. The figure is closely related to the Pappus chain.
Two of the semicircles are necessarily concave, with arbitrary diameters a and b; the third semicircle is convex, with diameter a+b. [1]
The area of the arbelos is equal to the area of a circle with diameter HA.
Proof: For the proof, reflect the arbelos over the line through the points B and C, and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters BA, AC) are subtracted from the area of the large circle (with diameter 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), the problem reduces to showing that . The length |BC| equals the sum of the lengths |BA| and |AC|, so this equation simplifies algebraically to the statement that . Thus the claim is that the length of the segment AH is the geometric mean of the lengths of the segments BA and AC. Now (see Figure) the triangle BHC, being inscribed in the semicircle, has a right angle at the point H (Euclid, Book III, Proposition 31), and consequently |HA| is indeed a "mean proportional" between |BA| and |AC| (Euclid, Book VI, Proposition 8, Porism). This proof approximates the ancient Greek argument; Harold P. Boas cites a paper of Roger B. Nelsen [3] who implemented the idea as the following proof without words. [4]
Let D and E be the points where the segments BH and CH intersect the semicircles AB and AC, respectively. The quadrilateral ADHE is actually a rectangle.
The line DE is tangent to semicircle BA at D and semicircle AC at E.
The altitude 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 parbelos is a figure similar to the arbelos, that uses parabola segments instead of half circles. A generalisation comprising both arbelos and parbelos is the f-belos, which uses a certain type of similar differentiable functions. [5]
In the Poincaré half-plane model of the hyperbolic plane, an arbelos models an ideal triangle.
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.
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.
In elementary geometry, two geometric objects are perpendicular if their intersection forms right angles at the point of intersection called a foot. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂. Perpendicular intersections can happen between two lines, between a line and a plane, and between two planes.
In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.
In mathematics, a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180°. It has only one line of symmetry.
In hyperbolic geometry, two lines are said to be ultraparallel if they do not intersect and are not limiting parallel.
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. If the arbelos is partitioned into two smaller regions by a line segment through the middle point of A, B, and C, perpendicular to line ABC, then each of the two twin circles lies within one of these two regions, tangent to its two semicircular sides and to the splitting segment.
In geometry, Archimedes' quadruplets are four congruent circles associated with an arbelos. Introduced by Frank Power in the summer of 1998, each have the same area as Archimedes' twin circles, making them Archimedean circles.
The salinon is a geometrical figure that consists of four semicircles. It was first introduced in the Book of Lemmas, a work attributed to Archimedes.
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 r denotes the radiius of any of the inner half circles, then the radius ρ of such an Archimedean circle is given by
The Book of Lemmas or Book of Assumptions is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions (lemmas) on circles.
In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex plane region bounded by one 180-degree circular arc and one 90-degree circular arc. It was the first curved figure to have its exact area calculated mathematically.
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.
In geometry, the Newton–Gauss line is the line joining the midpoints of the three diagonals of a complete quadrilateral.
The parbelos is a figure similar to the arbelos but instead of three half circles it uses three parabola segments. More precisely the parbelos consists of three parabola segments, that have a height that is one fourth of the width at their bases. The two smaller parabola segments are placed next to each other with their bases on a common line and the largest parabola is placed over the two smaller ones such that its width is the sum of the widths of the smaller ones.