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ISOTEL CURVE

Notion studied by Lepiney in 1909.
From the Greek iso "same" and têle "far away". |

The isotel curve of a plane curve (G_{0}) with respect to a point O is the locus of the points located at equal distance from O and (G_{0}),
in other words, the equidistance curve of (G_{0}) and
O.
If M describes (G_{0}_{0}), then the isotel curve is the locus of the intersection points between the perpendicular bisector of [OM] and the normal at _{0}M; therefore, it is also the locus of the centres of the circles passing by _{0}O and tangent to the curve (G_{0}); the isotel is thus none other than the curve the orthotomic of which is the initial curve, in other words, its "anti-orthotomic".
Since the orthotomic is the image of the pedal by a homothety with centre O and ratio 2, the isotel is none other than the image of the negative pedal by a homothety with centre O and ratio 1/2; the notion therefore became obsolete. |

Conclusion: up to homothety, isotel = negative pedal = orthocaustic.

Examples:

- the parabola is the isotel of its directrix with respect to its focus.

- a centred conic is the isotel of the directrix circle at one focus with respect to the other focus.

- the isotel curve of the ellipse with respect to its centre is the Talbot curve.

For other examples, see negative pedal.

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© Robert FERRÉOL 2017