A Geometrical Treatise On Conic Sections. With Numerous Examples ..

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Draw EQW parallel to OP, meeting the circle in W, and TP produced in B.
Also, draw QF parallel to PB, meeting the diameter PP' in V; then since ^P touches the circle at P, 104 CONIC SECTIONS.
CP\ {Prop. XXVI.) : CP^ GP\ .■.RQ.BW= PB\ {Euclid, III. 36) oiPV.BW= QV\ ButQV'iPV.P'Vi: 017 ■.PV. RW : PV . P'V :: GIT ovBW-.PV:: CD' Now, when the circle becomes the circle of curvature at P, the points B and Q move up to, and coincide with P, and the lines RW and P^ff become equal, while P' V becomes eq...ual to PP', or 2 CP.
B.ence, PH : 20P :: CD' -.OP', ■. PH.GP: 2GP^ :: 2017 : 20P\ . . PH .OP =2 GD\ Pkop, XXX.
If P Z7 be the diameter of the circle of curvature at the point P of the hyperbola, and PF be drawn at right angles to GD; then PU .PF=2GJ}\ CONIC SECTIONS. 105 Since the triangle PHU is similar to the triangle PFG, .-. PU : PH :: CP : PF, .-. PU . PF = PH : CP, = 2Giy\ {Prop. XXIX.) Prop. XXXI.
If PI be the chord of the circle of curvature through the focus of the hyperbola ; then PI.AG=^2CD\ Let 8'P meet CD in E; then since the triangles PIU and PFF&ve similar, .-.

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