|
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) : 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 : 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. ==Orthogonal circle== For a point P outside the circle, the power ''h'' equals ''R''2, the square of the radius ''R'' of a new circle centered on P that intersects the given circle at right angles, i.e., orthogonally (Figure 2). If the two circles meet at right angles at a point T, then radii drawn to T from P and from O, the center of the given circle, likewise meet at right angles (blue line segments in Figure 2). Therefore, the radius line segment of each circle is tangent to the other circle. These line segments form a right triangle with the line segment connecting O and P. Therefore, by the radical center of three circles. The point T can be constructed—and, thereby, the radius ''R'' and the power ''p'' found geometrically—by finding the intersection of the given circle with a semicircle (red in Figure 2) centered on the midpoint of O and P and passing through both points. By simple geometry, it can also be shown that the point Q is the inversion (geometry)#Circle inversion|inverse]] of P with respect to the given circle. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Power of a point」の詳細全文を読む スポンサード リンク
|