Casey's theorem

The set of Casey is a named after the Irish mathematician John Casey set of elementary geometry. It represents an extension of the set of Ptolemy and describes the behavior of the tangent sections of four circles in a specific configuration.

Summing up the Tangtenabschnitte of adjacent in the numbering circles as " tangent outsides " (black) and the non-adjacent as " tangent diagonals " (red ), so can also be formulated as the set:

The equation is referred to as Casey condition.

Extensions and applications

Leaving the radii of the circles tend to zero, so they go in the limit to points above on the circle and the tangent sections to the sides and diagonals of a quadrilateral tendon. The set of Ptolemy is thus obtained as a limiting case.

The set of Casey remains valid even if it is a degenerate circle, ie a point (radius zero) or a line ( infinite radius ), is. Therefore apply the following two sentences:

The reversal of the sentence of Casey is also true, ie it is the following sentence:

History

Within the western mathematics of sentence was first published by John Casey, however, was a special case of the theorem, in which the 4 circle tangent are also inscribed in a square, also in Japanese Mathematics in the Edo period ( Wasan ) known. He is survived, among other things, in the form of a Sangaku problem of 1874 from the Gunma prefecture and was already in the 1820s the mathematician Chochu Siraishi known.