70 The Three Sections, Tangencies, 



BP2 + c.d ; : m : n^ — (the ratio ~ being known in sign and 

 magnitude, and the rectangles a.b, c.d being also of known 

 magnitudes and signs). 



ANALYSIS. 



Suppose the points C and D on AP and BP such that 

 CA.AP = a.b, and DB.BP = c.d; then we have AP.CP : 

 BP. DP : : : mn. 



But if Q be the point in BP such that the angle QA right 

 to P = CP right to D, we have QP.DP = AP.CP; and 

 therefore it follows that QP : BP :: m : n. 



Hence if we draw PO parallel to AQ to cut AB in O^ we 

 have AO : BO : : m : n, and therefore the point O is known. 

 Moreover, the angle PO right to Q or D being equal Q,A 

 right to P or T), it is equal to CP right to T>, and therefore 

 OP is a tangent to the circle CDP. 



Now it is evident that if P' be any point whatever, and 

 that C and D' are taken on AP' and BP' so that CA.AP' = 

 CA.AP, and D'B.BP' = DB.BP, then will the circles PCD 

 and P'C'D' have AB as radical axis, and therefore (as O is a 

 point on this radical axis) the tangent OP equal to the tangent 

 from O to the circle P'C'D'; but the circle P'C'D' is known; 

 therefore OP is of kno\\Ti magnitude, and hence the locus of 

 P is a known circle having O for centre. 



COMPOSITION, 



Find O in AB such that AO : BO : : m : n; assume any 

 point P', and in the straight lines AP' and BP' find the points 

 C and D' such that CA.AP' = a.b, and D'B.BP' = c.d; 

 describe the circle P'C'D', and draw a tangent to it from O ; 

 with O as centre and this tangent as radius describe a circle ; 

 this circle will be the required locus. 



For let P be any point in its circumference, and C and D 

 the points in AP and BP such that CA.AP = CA.AP', and 

 DB.BP = D'B.BP'; and let Q be the point in which AQ 

 parallel to PO cuts PB. 



The straight line ABO is e^ddently the radical axis of the 

 circles PCD and P'C'D', and therefore OP is a tangent to the 

 circle PCD. Hence angle CP right to D being equal angle 

 PO right to D, it is equal QA right to D or P, and therefore 



