CHAP. IV.] 



OVAL CURVES. 



95 



PROPOSITION 7 PROBLEM. 



To draw a tangent to a given Oval, the foci and the ratio being 

 given : 



It is required to draw a tangent at C to the oval CNT, ratio m : n. 

 From B the less focus describe a circle as in (Prop. 3). Join CA, join 

 CB, and produce to D, then DC : CA ::m:n. Join AD and pro- 

 duce it. Bisect the angle DCA by the line CO. DO : OA : : 

 (DC : CA : :) m : n. Draw CK perpendicular to CO, and describe 

 the circle OCLK. Because OCK is a right angle, OCK = DCO + 

 KCB = OCA + ACK. But OCA = DCO .-. KCB = ACK and (6.3) 

 KA : KD : : CA : CD : : AO : OD .-. KA : KD : : AO : OD, and at any 

 point L in the circle, AL : DL : : AO : OD (6 F), and AL : DL : : n : m ; 

 and the circle is wholly without the oval, and can only touch it in the 

 two points C and P, where DB cuts the oval ; for suppose the circle 

 coincided with the oval at L, 



Join BL and produce to X, then AL : DL : : n : m ) m ^j. _ 

 And by Prop. 3 . . AL : XL ::n:m / ' 



but DL > XL (3.7), therefore the oval is within the circle. There- 

 fore draw ECF a tangent to the circle, and as it is without the circle 

 it is without the oval. 



XL 



