Жарты шеңберлермен шектелген жазық аймақ (Arbelos)

Plane region bounded by three semicircles

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. who implemented the idea as the following proof without words.

Let D and E be the points where the segments and intersect the semicircles AB and AC, respectively. The quadrilateral ADHE is actually a rectangle. Proof: ∠BDA, ∠BHC, and ∠AEC are right angles because they are inscribed in semicircles (by Thales's theorem). The quadrilateral ADHE therefore has three right angles, so it is a rectangle. Q. E. D.

The line DE is tangent to semicircle BA at D and semicircle AC at E.
Proof: Since ∠BDA is a right angle, ∠DBA equals minus ∠DAB. However, ∠DAH also equals minus ∠DAB (since ∠HAB is a right angle). Therefore triangles DBA and DAH are similar. Therefore ∠DIA equals ∠DOH, where I is the midpoint of and O is the midpoint of But ∠AOH is a straight line, so ∠DOH and ∠DOA are supplementary angles. Therefore the sum of ∠DIA and ∠DOA is π. ∠IAO is a right angle. The sum of the angles in any quadrilateral is 2π, so in quadrilateral IDOA, ∠IDO must be a right angle. But ADHE is a rectangle, so the midpoint O of (the rectangle's diagonal) is also the midpoint of (the rectangle's other diagonal). As I (defined as the midpoint of ) is the center of semicircle BA, and angle ∠IDE is a right angle, then DE is tangent to semicircle BA at D. By analogous reasoning DE is tangent to semicircle AC at E. Q. E. D.

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. 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.