6.2. † 47. 1. 6.2. thereto, the Rectangle under A D and D C, together with the Square of EC, thall be equal to the Square of E D. But EC is equal to E B ; wherefore the Recangle under A D and DC, together with the Square of E B, is equal to the Square of ED. But the Square of ED is + equal to the Squares of E B and B D, tor the Angle E B D is a Right Angle: Therefore the Rectangle under A D and DC, together with the Square of E B, is equal to the Squares of E B and B D; and if the common Square of E B be taken away, the Rectangle under AD and DC remaining, will be equal to the Square of the Tangent Line BD. * Now, let DC A not pafs through the Centre of the 1of this. Circle ABC; and find the Centre E thereof, and draw E F perpendicular to A C, and join E B, E C, ED. Therefore EFD is a Right Angle. And becaufe a Right Line E F, drawn thro' the Centre, cuts a Right Line A C, not drawn thro' the Centre, at Right 3 of this. Angles, it will bifect the fame; and fo A F is equal to F C. Again, fince the Right Line A C is bifected in F, and CD is added thereto, the Rectangle under A D and D C, together with the Square of F C, will be equal to the Square of F D. And if the common Square of EF be added, then the Rectangle under AD and DC, together with the Squares of FC and FE, is equal to the Squares of DF and FE. But the Square of DE is equal to the Squares of D F and FE for the Angle EFD is a Right one; and the Square of CE is equal to the Squares of CF and + FE. Therefore the Rectangle under AD and DC, together with the Square of CE, is equal to the Square of ED; but GE is equal to E B. Wherefore the Rectangle under AD and DC, together with the Square of E B, is equal to the Square of E D. But the Squares of E B and B D are equal to the Square of ED; fince the Angle EBD is a Right one. Wherefore the Rectangle under A D and DC, together with the Square of E B, is equal to the Squares of EB and B D. And if the common Square of EB bè taken away, the Rectangle under AD and D C, remaining, will be equal to the Square of DB. Therefore, if any Point be taken without a Circle, and from that Point two Right Lines fall to the Circle, one of which cuts the Circle, and the other touches it; the Rectangle contained † 47. I. under under the whole Secant Line, and its Part between the Convexity of the Circle and the affumed Part, will be equal to the Square of the Tangent Line; which was to be demonftrated. If fome Point be taken without a Circle, and two Right Lines be drawn from it to the Circle, fo that one cuts it, and the other falls upon it; and if the Rectangle under the whole Secant Line, and the Part thereof, without the Circle, be equal to the Square of the Line falling upon the Circle then this laft Line will touch the Circle. Nou LET fome Point D be affumed without the Circle 17 of this. of the Cir- 11 of this. By Hyp Then the Angle FED is a Right Angle. And +18 of it is. becaufe D E touches the Circle A B C, and DCA cuts it, the Rectangle under A D and D C will be equal to the Square of D E. But the Rectangle under AD and DC is equal to the Square of D B. Wherefore the Square of D E fhall be equal to the Square of D B. And fo the Line DE will be equal to the Line D B. But E F is equal to FB: Therefore the two Sides DE, EF, are equal to the two Sides DB, BF, and the Bafe FD is common. Where fore the Angle DEF is equal to the Angle DBF; 3. 1. But DEF is a Right Angle; wherefore, DBF is alfo a Right Angle, and F B produced is a Diameter. But a Right Line drawn at Right Angles, on the End of the Diameter of a Circle, touches the Circle; therefore BD neceffarily touches the Circle We prove this in the fame manner, if the Centre of the Circle be in the Right Line CA. If, therefore, any Point be af fumed without a Circle, and two Right Lines be drawn from it to the Circle, fo that one cuts it, and the other falls upon it; and if the Rectangle under the whole Secant Line, and the Part thereof, without the Circle, be equal to the Square of the Line falling upon the Circle; then this laft Line will touch the Circle; which was to be demonftrated. Caroll. Hence, if from any Point, without a Circle, feveral Right Lines A B, A C, are drawn, cutting the Circle, the Rectangles comprehended under the whole Lines A B, A C, and their external Parts A E, A F, are equal between themselves. For, if the Tangent A D be drawn, the Rectangle under BA and AD is equal to the Square of A D; and the Rectangle under C A and A F is equal to the fame Square of AD: Therefore the Rectangles fhall be equal. The END of the THIRD BOOK. EUCLI D's |