What is "power of a point"

Power \Pow"er\, n. [OE. pouer, poer, OF. poeir, pooir, F. pouvoir, n. & v., fr. LL. potere, for L. posse, potesse, to be able, to have power. See Possible, Potent, and cf. Posse comitatus.]

1. Ability to act, regarded as latent or inherent; the faculty of doing or performing something; capacity for action or performance; capability of producing an effect, whether physical or moral: potency; might; as, a man of great power; the power of capillary attraction; money gives power. ``One next himself in power, and next in crime.''
   --Milton.

2. Ability, regarded as put forth or exerted; strength, force, or energy in action; as, the power of steam in moving an engine; the power of truth, or of argument, in producing conviction; the power of enthusiasm. ``The power of fancy.''
   --Shak.

3. Capacity of undergoing or suffering; fitness to be acted upon; susceptibility; -- called also passive power; as, great power of endurance.

   Power, then, is active and passive; faculty is active power or capacity; capacity is passive power.
   --Sir W. Hamilton.

4. The exercise of a faculty; the employment of strength; the exercise of any kind of control; influence; dominion; sway; command; government.

   Power is no blessing in itself but when it is employed to protect the innocent.
   --Swift.

5. The agent exercising an ability to act; an individual invested with authority; an institution, or government, which exercises control; as, the great powers of Europe; hence, often, a superhuman agent; a spirit; a divinity. ``The powers of darkness.''
   --Milton.

   And the powers of the heavens shall be shaken.
   --Matt. xxiv. 29.

6. A military or naval force; an army or navy; a great host.
   --Spenser.

   Never such a power . . . Was levied in the body of a land.
   --Shak.

7. A large quantity; a great number; as, a power o? good things. [Colloq.]
   --Richardson.

8. (Mech.)

   1. The rate at which mechanical energy is exerted or mechanical work performed, as by an engine or other machine, or an animal, working continuously; as, an engine of twenty horse power.

      Note: The English unit of power used most commonly is the horse power. See Horse power.

   2. A mechanical agent; that from which useful mechanical energy is derived; as, water power; steam power; hand power, etc.

   3. Applied force; force producing motion or pressure; as, the power applied at one and of a lever to lift a weight at the other end.

      Note: This use in mechanics, of power as a synonym for force, is improper and is becoming obsolete.

   4. A machine acted upon by an animal, and serving as a motor to drive other machinery; as, a dog power.

      Note: Power is used adjectively, denoting, driven, or adapted to be driven, by machinery, and not actuated directly by the hand or foot; as, a power lathe; a power loom; a power press.

9. (Math.) The product arising from the multiplication of a number into itself; as, a square is the second power, and a cube is third power, of a number.

10. (Metaph.) Mental or moral ability to act; one of the faculties which are possessed by the mind or soul; as, the power of thinking, reasoning, judging, willing, fearing, hoping, etc.
    --I. Watts.

    The guiltiness of my mind, the sudden surprise of my powers, drove the grossness . . . into a received belief.
    --Shak.

11. (Optics) The degree to which a lens, mirror, or any optical instrument, magnifies; in the telescope, and usually in the microscope, the number of times it multiplies, or augments, the apparent diameter of an object; sometimes, in microscopes, the number of times it multiplies the apparent surface.

12. (Law) An authority enabling a person to dispose of an interest vested either in himself or in another person; ownership by appointment.
    --Wharton.

13. Hence, vested authority to act in a given case; as, the business was referred to a committee with power.

    Note: Power may be predicated of inanimate agents, like the winds and waves, electricity and magnetism, gravitation, etc., or of animal and intelligent beings; and when predicated of these beings, it may indicate physical, mental, or moral ability or capacity.

    Mechanical powers. See under Mechanical.

    Power loom, or Power press. See Def. 8 (d), note.

    Power of attorney. See under Attorney.

    Power of a point (relative to a given curve) (Geom.), the result of substituting the co["o]rdinates of any point in that expression which being put equal to zero forms the equation of the curve; as, x^ 2 + y^ 2 - 100 is the power of the point x, y, relative to the circle x^ 2 + y^ 2 - 100 = 0.

In elementary plane geometry, the power of a point is a real number h that reflects the relative distance of a given point from a given circle. Specifically, the power of a point P with respect to a circle O of radius r is defined by (Figure 1)

h = s − r, 

where s is the distance between P and the center O of the circle. By this definition, points inside the circle have negative power, points outside have positive power, and points on the circle have zero power. For external points, the power equals the square of the length of a tangent from the point to the circle. The power of a point is also known as the point's circle power or the power of a circle with respect to the point.

The power of point P (see in Figure 1) can be defined equivalently as the product of distances from the point P to the two intersection points of any ray emanating from P. For example, in Figure 1, a ray emanating from P intersects the circle in two points, M and N, whereas a tangent ray intersects the circle in one point T; the horizontal ray from P intersects the circle at A and B, the endpoints of the diameter. Their respective products of distances are equal to each other and to the power of point P in that circle

$$\mathbf{\overline{PT}}^2 = \mathbf{\overline{PM}} \times \mathbf{\overline{PN}} = \mathbf{\overline{PA}} \times \mathbf{\overline{PB}} = (s - r) \times (s + r) = s^2 - r^2 = h.$$

This equality is sometimes known as the "secant-tangent theorem", "intersecting chords theorem", or the "power-of-a-point theorem".

The power of a point is used in many geometrical definitions and proofs. For example, the radical axis of two given circles is the straight line consisting of points that have equal power to both circles. For each point on this line, there is a unique circle centered on that point that intersects both given circles orthogonally; equivalently, tangents of equal length can be drawn from that point to both given circles. Similarly, the radical center of three circles is the unique point with equal power to all three circles. There exists a unique circle, centered on the radical center, that intersects all three given circles orthogonally, equivalently, tangents drawn from the radical center to all three circles have equal length. The power diagram of a set of circles divides the plane into regions within which the circle minimizing the power is constant.

More generally, French mathematician Edmond Laguerre defined the power of a point with respect to any algebraic curve in a similar way.