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 C of radius r is defined (Figure 1)
where s is the distance from P to 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 radius of a circle centered on the given point that intersects the given circle orthogonally, i.e., at right angles (Figure 2). 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.
More generally, Laguerre defined the power of a point with respect to any algebraic curve in a similar way.
The power of a point 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
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.
Read more about Power Of A Point: Orthogonal Circle, Theorems, Darboux Product, Laguerre's Theorem
Other articles related to "power of a point, point":
... Laguerre defined the power of a point P with respect to an algebraic curve of degree n to be the product of the distances from the point to the intersections of a circle through the point ... In the case when the algebraic curve is a circle this is not quite the same as the power of a point with respect to a circle defined in the rest of this article, but ...
Famous quotes containing the words power of, point and/or power:
“Many a reformer perishes in his removal of rubbish,—and that makes the offensiveness of the class. They are partial; they are not equal to the work they pretend. They lose their way; in the assault on the kingdom of darkness, they expend all their energy on some accidental evil, and lose their sanity and power of benefit.”
—Ralph Waldo Emerson (1803–1882)
“The point is, that the function of the novel seems to be changing; it has become an outpost of journalism; we read novels for information about areas of life we don’t know—Nigeria, South Africa, the American army, a coal-mining village, coteries in Chelsea, etc. We read to find out what is going on. One novel in five hundred or a thousand has the quality a novel should have to make it a novel—the quality of philosophy.”
—Doris Lessing (b. 1919)
“We have no higher life that is really apart from other people. It is by imagining them that our personality is built up; to be without the power of imagining them is to be a low-grade idiot.”
—Charles Horton Cooley (1864–1929)