Power line

The power line ( power line , chordale ) of two circles is understood to mean the geometric location (the set ) of all points whose power corresponds to the two circles. If the circles are given by their centers and as well as their radii and , then the power line consists exactly of the points for which ${\displaystyle M_{1}}$ ${\displaystyle M_{2}}$ ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$ ${\displaystyle P}$

${\displaystyle {\overline {M_{1}P}}^{2}-{r_{1}}^{2}={\overline {M_{2}P}}^{2}-{r_{2}}^{2}}$

applies. The power line is only defined if the given circles are not concentric, i.e. do not have a coinciding center.

properties

• The power line of two circles runs perpendicular to the line connecting the two circle centers.

For two intersecting circles (case 3) the power line goes through the two intersection points. If the two circles touch each other (case 2 and case 4), the power line coincides with the common tangent.

Three circles (black) with power lines (blue); the orange drawn circle intersects the given circles at right angles.

• The distances of the power lines from the two circle centers are given by:

${\displaystyle d_{1}={\frac {d^{2}+{r_{1}}^{2}-{r_{2}}^{2}}{2d}}}$

${\displaystyle d_{2}={\frac {d^{2}+{r_{2}}^{2}-{r_{1}}^{2}}{2d}}}$

And denote the radii; stands for the distance between the centers. ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$ ${\displaystyle d={\overline {M_{1}M_{2}}}}$

• For the points of the power line that lie outside the given circles, the tangent segments on both circles are of equal length.

• The power line of two circles is the set of the centers of all circles that intersect the given circles at right angles.

• If there are three circles, among which no two are concentric, there are three power lines (one to two circles each). If the centers of the given circles do not lie on a straight line, the power lines intersect at a point ( radical center ), namely in the center of the circle that intersects the given circles at right angles ( radical circle ).

literature

• Max Koecher , Aloys Krieg : level geometry . Springer, 2013, ISBN 9783662068090 , pp. 138-140
• Harald Scheid, Wolfgang Schwarz: Elements of geometry . Springer, 2016, ISBN 9783662503232 , pp. 156–158

Web links

• Eric W. Weisstein : Radical line . In: MathWorld (English).