Generalization of the triangle and Ptolemy inequalities (Q1336199)
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scientific article; zbMATH DE number 663725
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Generalization of the triangle and Ptolemy inequalities |
scientific article; zbMATH DE number 663725 |
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Generalization of the triangle and Ptolemy inequalities (English)
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7 November 1994
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The triangle inequality \(| A_ 1 A_ 2 | + | A_ 2 A_ 3 | \geq | A_ 1 A_ 3 |\) can be written in the form \[ 1/ | A_ 1 A_ 2 | \cdot | A_ 1 A_ 3| - 1/| A_ 2 A_ 1 | \cdot | A_ 2 A_ 3 | + 1/| A_ 3 A_ 1 | \cdot | A_ 3 A_ 2 | \geq 0. \] In this form it can be generalized to an alternating inequality for any odd-sided polygon that can be inscribed in a circle; there is equality for an even-sided polygon. Use of inversion leads to a generalization of Ptolemy's theorem.
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triangle inequality
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Ptolemy's theorem
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