is a right angle). Therefore triangles and are similar. Therefore equals , where is the midpoint of and is the midpoint of . But is a straight line, so and are supplementary angles. Therefore the sum of and is π. is a right angle. The sum of the angles in any quadrilateral is 2π, so in quadrilateral , must be a right angle. But is a rectangle, so the midpoint of (the rectangle's diagonal) is also the midpoint of (the rectangle's other diagonal). As (defined as the midpoint of ) is the center of semicircle , and angle is a right angle, then is tangent to semicircle at . By analogous reasoning is tangent to semicircle at . Q.E.D. Archimedes' circles The altitude 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. Variations and generalisations 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. Etymology 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. See also Archimedes' quadruplets Bankoff circle Schoch circles Schoch line Woo circles Pappus chain Salinon References Bibliography American Mathematical Monthly, 120 (2013), 929-935. External links