GEOMETRICAL PORISMS. 121 



of the given lines. Let AG, GK be fo drawn as to form a 

 triangle AGK, fimilar to the given triangle def, and having its 

 angles placed upon the given lines, in the given order. Through 



A, G, any two of its angles, and C, the interfedtion of the lines 

 upon which they are placed, defcribe a circle j through G, K, and 



B, the interfedlion of CG, AK, let another circle be defcribed, 

 meeting the former in H, the point to be found, which will alfo 

 be in a circle pafling through K, and touching CA at A. 



The demonftration follows eafily from the preceding analyfis. 



Cor. r. The Unes HD, HE, HF contain given angles, and 

 have to each other the fame ratios, with the given lines HA, 

 HG, HK. 



CoR. 2. The lines AD, GE, KF have alfo to each other the 

 given ratios of HA, HG, HK. 



PROP. IX. THEOREM, Fig. 14. PI. III. 



Let Ea, E^, Fc, Gd, Uc. be any number of ftraight lines 

 given by pofition. Let P be a given point. Through P, 

 and E, the interfe(Elion of any two of the given lines, let a 

 circle be defcribed to meet them in A and B ; through P, B, 

 and H, the interfedlion of B^, with one of the remaining 

 lines, let a circle be defcribed to meet that line in C. Through 

 P, C, and K, the interfedlion of Cc, with one of the remain- 

 ing lines, let a circle be defcribed to meet that line in D, 

 and fo on if there be more lines. Join AB, BC, CD, ijc, 

 DA. The recSlilineal figure ABCD, Is^c. is given in fpecies. 



Take a, a given point in EA, through P. E, a, defcribe a 

 circle to meet EB in b, through "P, H, b, defcribe a circle to meet 

 HC in c, through P, K, c, defcribe a circle to meet KD in d, and 

 fo on if there be more lines. Join Fa, PA, alfo PB, P3, PC, 



P Tc, 



