Parbelos

Parbelos with parallelogram outer tips and inner tip ${\displaystyle BM_{2}MM_{1}}$ ${\displaystyle A,C}$ ${\displaystyle B}$
${\displaystyle F_{\text{Parbelos}}={\frac {4}{3}}F_{\text{Parallogramm}}}$
${\displaystyle |{\widehat {AC}}|=|{\widehat {AB}}|+|{\widehat {BC}}|}$

Nested Parbelos, the two gray semicircles are congruent

Parbelos with tangent rectangle The tangent to a parabolic arc with an intersection ${\displaystyle BDEF}$
${\displaystyle DF}$ ${\displaystyle {\widehat {AC}}}$
${\displaystyle G=DF\cap {\widehat {AC}}=DF\cap n={\widehat {AC}}\cap n}$

The Parbelos is a figure similar to the Arbelos , in which parabolic segments are used instead of the semicircles . It applies to all parabolic segments that their height corresponds to a quarter of their width at the base. This means that the parabolic segments are created by separating a parabola along the parallels to the guideline through the focal point .

Some of the characteristics of the Parbelos are similar or even the same as those of the Arbelos. For example, as with the Arbelos, the following two statements apply:

• The length of the outer parabolic arch corresponds to the sum of the lengths of the two inner parabolic arches.
• In the case of a nested Parbelos, i.e. if one again forms a Parbelos with the two inner parabolic segments, the two inner parabolas of the new Parbelos constructions, which are located at the inner tip of the outer Parbelos, are the same size.

The square formed by the inner tip and the centers of the three parabolic arcs is a parallelogram and the following applies to its area: ${\displaystyle B}$ ${\displaystyle M_{2},M,M_{1}}$ ${\displaystyle BM_{2}MM_{1}}$

${\displaystyle F_{\text{Parallelogram}}={\frac {3}{4}}F_{\text{Parbelos}}}$

The four tangents at the three points of the Parbelos form a rectangle, the so-called tangent rectangle. Its circumference cuts the base of the outer parabolic segment in its center and thus runs through the focal point of the outer parabola. One of the diagonals of the tangent rectangle lies on a tangent of the outer parabola and the point of contact is identical to the intersection of the diagonals with the perpendicular to the base at the inner tip. The following formula applies to the area of ​​the tangent rectangle:

${\displaystyle F_{\text{Rechteck}}={\frac {3}{2}}F_{\text{Parbelos}}}$

See also

• Parabolic arbelos

literature

• Jonathan Sondow: `` The Parbelos, a Parabolic Analog of the Arbelos ''. In: The American Mathematical Monthly , Volume 120, No. 10 (December 2013), pp. 929-935 ( JSTOR )
• Michał Różański, Alicja Samulewicz, Marcin Szweda, Roman Wituła: Variations on the arbelos . In: Journal of Applied Mathematics and Computational Mechanics , Volume 16, Issue 2, 2017, pp. 123-133 ( digitized version )
• Emmanuel Tsukerman: Solution of Sondow's Problem: A Synthetic Proof of the Tangency Property of the Parbelos . In: The American Mathematical Monthly , Volume 121, No. 5 (May 2014), pp. 438-443
• Antonio M. Oller-Marcén: The f-belos . In: Forum Geometricorum , 13 (2013), pp. 103–111 ( online copy )

Web links

Commons : Parbelos  - collection of images, videos and audio files

• Viktorija Ternar: Arbelos, parabelos in f-belos - Master thesis, University of Maribor, 2015

Individual evidence

1. ↑ ^{a b} Michał Różański, Alicja Samulewicz, Marcin Szweda, Roman Wituła: "Variations on the arbelos". In: Journal of Applied Mathematics and Computational Mechanics , Volume 16, Issue 2, 2017, pp. 123-133 ( digitized version )
2. ↑ Jonathan Sondow: The Parbelos, a Parabolic Analog of the Arbelos . In: The American Mathematical Monthly , Volume 120, No. 10 (December 2013), pp. 929-935 ( JSTOR )