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