60 The Three Sections, Tangencies, 



BA as diameter cuts AO^ we have 2.0 A. GO = — OA^ — 



OB^ + AB^ . And from these two we get^ by adding equal 

 quantities, the relation 20A.GM = AB^ — AM^ . 



Similarly if in OA we suppose ON equal to OC and the 

 like direction in respect to direction OD which the direction 

 OC has to OF, then will DN = FC, and will AN be of known 

 magnitude. And it is evident that if H be the point in which 

 the circle having CA as diameter cuts AO, we have in like 

 manner the relation 20A.HN = CA^ — AN^ . 



But the ratio of the known quantities AB^ — AM^ and CA^ 

 — AN2 is known : .-.it follows that the ratio of GM to HN 

 which is the same with it is known. 



Now if in MN we suppose NL so taken that MA has to 

 NL the known ratio which GM has to HN ; then GA has to 

 HL the same ratio. 



Hence if in AO we suppose KA = HL and in like 

 direction to it, then GA has to KA a known ratio, and .-. 

 the point K must be in the circumference of a known circle 

 AKP passing through A and having its diameter AQ in AB. 



Again since KA = HL, we have KH = the known 

 magnitude AL. And the angles CHK, QKH, being right, 

 it follows that QT the perpendicular from Q, on CH is equal 

 to KH, and that CT is tangent to the circle having Q, as 

 centre and QT as radius ; but this circle is known ; there- 

 fore the tangent CT is known, as also the other point H in 

 which it cuts the circle on AC as diameter; and therefore 

 AHO is known in position. 



And, since AHO is known in position, the point N is 

 known; and .-. as OC = ON, the point O is known, and 

 hence the circle DEF. 



COMPOSITION. 



Through the centre A draw any radius AD' ; in B'A take 

 D'M' = radius B, and D'N' = radius C ; in M'N'A find the 

 point L' such that M'A shall have to N'L the ratio which 

 AB^ — AM'2 has to CA.^ — AN'^ (taking signs into account) ; 

 in AB take AQ so that AB : AQ : : M'A : N'L (taking 

 note of signs) ; on AC and AQ, as diameters describe circles ; 

 with Q as centre and a radius equal to AL' describe a circle, 

 and draw CT a tangent to it from C ; through the other point 

 H in which CT cuts the circle on AC as diameter, draw the 

 straight line AH to cut the circle on AQ as diameter in K ; 

 make HL = KA (and in the same direction) ; make in the 



