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Homediametral points

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# 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. 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.

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

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

## Mathematics Subject Classification

51N20*no label found*51M04

*no label found*

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