VII. 3] POWER OF A POINT 213 In hyperbolic geometry tan is replaced by i tanh. This product may be called the power of the point O with respect to the circle. It is positive if O is outside, negative if O is inside the circle. In the former case, if t is the length of the tangent from 0 to the circle, the power of O is equal to tan2it. 3. Power of a point with respect to an equidistant curve. (1) Let the secant cut one branch of the curve in P, Q, i.e. in hyperbolic geometry the secant does not cut the axis of the curve, in elliptic geometry neither of the finite segments OP, OQ cuts the axis. Let M be the middle point of PQ, and draw MN 1 the axis; then MN is also 1 PQ. Draw OH the axis. Let OH =d, MN =x, OP =r, OQ =r', so that OM = {(r' +r), PM = } (r′ − r). Then, from the trirectangular quadrilaterals OHNM, PKNM, (2) Let the secant cut both branches of the curve, i.e. the point of intersection A with the axis is real and one of the segments OP, OQ cuts the axis. Let OAN = 0, ON =d, OP =r, OQ =r', so that Note. Figs. 105 and 106 have been drawn for the case of hyperbolic geometry. In elliptic geometry the equidistant-curve is convex towards the axis. In Fig. 105, in this case, either OH <PK or O lies between P and Q. If O is the same point in the VII. 3] POWER OF A POINT 215 two figures, the values of tan r/tan r' and tan r tan r', respectively for the secant which cuts and the secant which does not cut the axis, are equal. Hence, if a variable line through a fixed point O cuts a circle in P, Q and its axis in A, either the ratio or the product of the tangents of half the segments OP, OQ is constant, according as (1) one, or (2) both or neither of the segments contains the point A. If OT is a tangent to the curve, the constant is equal to tan2¿OT, and is called the power of the point O with respect to the circle. The two cases are simply explained in elliptic geometry. Let the dotted circle AA' represent the axis of the circle, which is P A: FIG. 107. represented in the diagram by a pair of circles. The secant cuts the two circles in P, P'; Q, Q'; and the axis in A, A'. These pairs of course represent single points. AA'=PP' =QQ' = π ; therefore OQ' = T - OQ. tan OP Therefore tan OP tan 10Q=tan {OP cot {OQ' tan 10Q' 4. Reciprocally, if P is a variable point on a fixed line PN, and the tangents PT, PT' from P to a fixed circle make angles 0, 0' with PN, we have in Fig. 108, VII. 4] POWER OF A LINE 217 This result is true also in euclidean geometry, the constant reducing to (d - a)/(d+a). For an equidistant-curve, let the line cut the axis in N at an angle a (Fig. 109). Then, and ' being taken positively, TPL=T'PL = ! ( 0 + 0′), NPL= 1 (0 − ('), cos a = cos x sin 1 (0 – 0′), X cos a = cos x sin { (0 + 0′) ; tan 10 whence, as before, is constant. tan 10 If the angles 0, 0' are measured in the same sense, then for 'we must put π-0', and we have tan 10 tan 10′ = const. If, the angles e, e' being measured in the same sense, both or neither of them contains the line joining P to the centre, then we have (Fig. 110) LPC = }} (0′ + π + 0) = 1⁄2); + 3 (0′ + 0), TPC = {TPT' = }} (π − 0′ + 0) — — — § ( 0′ – 0). · = |