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.
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 with the curve ... 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 differs from it by a ...
Famous quotes containing the words power of, point and/or power:
“If all power is in the people, if there is no higher law than their will, and if by counting their votes, their will may be ascertainedthen the people may entrust all their power to anyone, and the power of the pretender and the usurper is then legitimate. It is not to be challenged since it came originally from the sovereign people.”
—Walter Lippmann (18891974)
“Point me out the happy man and I will point you out either egotism, selfishness, evilor else an absolute ignorance.”
—Graham Greene (19041991)
“Of a truth, Knowledge is power, but it is a power reined by scruple, having a conscience of what must be and what may be; whereas Ignorance is a blind giant who, let him but wax unbound, would make it a sport to seize the pillars that hold up the long- wrought fabric of human good, and turn all the places of joy as dark as a buried Babylon.”
—George Eliot [Mary Ann (or Marian)