Power (geometry)

In geometry, the term power refers to a special measure introduced by Jakob Steiner in 1826 for how far outside or inside a circle a point is. The power of a point with respect to a circle with center and radius is given by the arithmetic expression ${\displaystyle P}$ ${\displaystyle M}$ ${\displaystyle r}$

${\displaystyle {\overline {MP}}^{2}-r^{2}.}$

There are three different cases:

• If the point is outside the circle, then its power is positive. If and are the points of intersection of any straight line through with the circle, then the power of according to the law of secants is equal . This statement is also correct if and coincide ( secant-tangent theorem ). From the Pythagorean theorem it follows that for a point P outside the circle, the power corresponds to the square of the length of a tangent segment (i.e. with or in the drawing ). ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle {\overline {PX}}\cdot {\overline {PY}}}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle {\overline {B_{1}P}}^{2}}$ ${\displaystyle {\overline {B_{2}P}}^{2}}$
• Points that lie on the circle have the power of 0.
• The power is negative for points inside the circle. If and are the points of intersection of an arbitrary straight line through with the circle, then the power of according to the chord law is equal . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle -{\overline {PX}}\cdot {\overline {PY}}}$

Related terms

• Power line

literature

• Jakob Steiner: Some geometrical considerations . In: Journal for pure and applied mathematics , Volume 1, 1826, pp. 161-184
• Jacob Steiner, CF Geiser, H. Schröter: Jacob Steiner's lectures on synthetic geometry. First part: The theory of conic sections in elementary representation . Teubner, 1867, p. 1-3 ( books.google.com ).
• Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 28-34

Web links

Commons : Power of a point  - collection of images, videos and audio files

• Florian Modler: Excursus: Power of a circle on matheplanet.de
• Eric W. Weisstein : Circle Power . In: MathWorld (English).
• Jacob Steiner and the Power of a Point at Convergence