# 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 DiametralPoints 2013-03-22 18:32:14 2013-03-22 18:32:14 pahio (2872) pahio (2872) 5 pahio (2872) Definition msc 51N20 msc 51M04 Antipodal diametral diametral circle