# power of point

###### Theorem.

If a secant of the circle is drawn through a point ($P$), then the product^{} of the line segments^{} on the secant between the point and the perimeter^{} of the circle is on the direction of the secant. The product is called the power of the point with respect to the circle.

Proof. Let $PA$ and $PB$ be the segments of a secant and $P{A}^{\prime}$ and $P{B}^{\prime}$ the segments of another secant. Then the triangles^{} $PA{B}^{\prime}$ and $P{A}^{\prime}B$ are similar^{} since they have equal angles, namely the central angles^{} $\mathrm{\angle}AP{B}^{\prime}$ and $\mathrm{\angle}BP{A}^{\prime}$ and the inscribed angles $\mathrm{\angle}PA{B}^{\prime}$ and $\mathrm{\angle}P{A}^{\prime}B$. Thus we have the proportion (http://planetmath.org/ProportionEquation)

$$\frac{PA}{P{A}^{\prime}}=\frac{P{B}^{\prime}}{PB},$$ |

which implies the asserted equation

$$PA\cdot PB=P{A}^{\prime}\cdot P{B}^{\prime}.$$ |

Notes. If the point $P$ is outside a circle, then value of the power of the point with respect to the circle is equal to the square of the limited tangent of the circle from $P$; this square (http://planetmath.org/SquareOfANumber) may be considered as the limit case of the power of point where the both intersection^{} points of the secant with the circle coincide. Another of the notion power of point is got when the line through $P$ does not intersect the circle; we can think that then the intersecting points are imaginary; also now the product of the “imaginary line segments” is the same.

Denote by ${p}^{2}$ the power of the point $P:=(a,b)$ with respect to circle

$$K(x,y):={(x-{x}_{0})}^{2}+{(y-{y}_{0})}^{2}-{r}^{2}=0.$$ |

Then, by the Pythagorean theorem^{}, we obtain

${p}^{2}={(a-{x}_{0})}^{2}+{(b-{y}_{0})}^{2}-{r}^{2}$ | (1) |

if $P$ is outside the circle and

${p}^{2}={r}^{2}-({(a-{x}_{0})}^{2}+{(b-{y}_{0})}^{2})$ | (2) |

if $P$ is inside of the circle. If in the latter case, we change the definition of the power of point to be the negative value $-{p}^{2}$ for a point inside the circle, then in both cases the power of the point $(a,b)$ is equal to

$$K(a,b)\equiv {(a-{x}_{0})}^{2}+{(b-{y}_{0})}^{2}-{r}^{2}.$$ |

Title | power of point |
---|---|

Canonical name | PowerOfPoint |

Date of creation | 2013-03-22 15:07:02 |

Last modified on | 2013-03-22 15:07:02 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 15 |

Author | PrimeFan (13766) |

Entry type | Theorem |

Classification | msc 51M99 |

Synonym | power of the point |

Synonym | power of a point |

Related topic | InversionOfPlane |

Related topic | VolumeOfSphericalCapAndSphericalSector |