# diametral points

Two points ${P}_{1}$ and ${P}_{2}$ on the circumference^{} of a circle (or on a sphere) are diametral, if the line segment^{} ${P}_{1}{P}_{2}$ connecting them passes through the centre of the circle (resp. the sphere), i.e. is a diametre (http://planetmath.org/Diameter^{}). Equivalently, the shortest distance^{} of the diametral points ${P}_{1}$ and ${P}_{2}$ on the circle is maximal on the circle (resp. on the sphere), namely a half of the perimetre (http://planetmath.org/Perimeter).

It’s easily justified that a point of a circle (resp. a sphere) has exactly one diametral point.

A circle $c$ is a diametral circle of a given circle ${c}_{0}$, if $c$ intersects ${c}_{0}$ diametrically, i.e. in two diametral points of ${c}_{0}$.

If the equation of ${c}_{0}$ is ${(x-{x}_{0})}^{2}+{(y-{y}_{0})}^{2}={r}^{2}$ and $(a,b)$ is inside ${c}_{0}$, then the equation of the diametral circle $c$ with centre $(a,b)$ is given by

$${(x-a)}^{2}+{(y-b)}^{2}={r}^{2}-{({x}_{0}-a)}^{2}-{({y}_{0}-b)}^{2}.$$ |

Title | diametral points |
---|---|

Canonical name | DiametralPoints |

Date of creation | 2013-03-22 18:32:14 |

Last modified on | 2013-03-22 18:32:14 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 51N20 |

Classification | msc 51M04 |

Related topic | Antipodal |

Defines | diametral |

Defines | diametral circle |