GEOMETRICAL PORISMS. 153 



the triangles EHD, GHA are fimilar. Now the triangle HEF 

 was proved fimilar to HGK. Therefore the quadrilateral HDEF 

 is fimilar to HAGK, and the angle DEF is equal to AGK ; alfo 

 DE is to EF as AG to GK ; therefore the triangle DEF is fimi- 

 lar to AGK or to def^ as was required. 



PROP. XVII. PROBLEM, Fig. 21. PI. IV. 



A and B are two given points in the circumference of a gi- 

 ven circle. C and D are two given points in ftraight lines 

 CE, DE given by polition. It is required to infledl AF, BF 

 to the given circumference, meeting the given lines in G 

 and H, fo that the redangle CG, DH may be equal to a 

 given fpace. 



Because A and B are giv^n points in the circumference of a 

 given circle^ and D is a given point in a line DE given by pofi- 

 tion, a line LM, and a point M in it, both given by pofition, 

 may be found, (prop. 10.), fo that BF, AF being infleded to 

 any point in the circumference, meeting the given line DE in 

 H ; and the line LM, which may be found in N, the ratio of 

 DH to MN, may be given. 



Suppose the line ML found, fo that MN may be equal to DH, 

 then the redlangle MN, CG is equal to DH, CG, which by hy- 

 pothefis is given. Now A is a given; point, and C, M are given 

 points in ftraight lines given by polition. Therefore the pro- 

 blem is now reduced to the 12th propolition of this paper. 



Construction. Join B and D, the given point, in the line 

 whofe fegment is to be intercepted by BF. Let BD meet the 

 circle in K ; join AK, and take AM equal to BD. Through the 

 points D, M, K defcribe a circle cutting DE in L, and AK in 

 M. Join LM, and from the point A (by prop, 12.) draw a 

 ftraight Hne to meet CE in G, and LM in N, fo that the rec- 

 tangle 



