GEOMETRICAL PORISMS. 113 



QX CE, and PX AD — Qx AE, will be equal to PX AB— Qx AC, 

 that is, by conflrucftion to tlie given fpace. 



LEMMA, Fig. 9. PL II. 



If circles be defcribed through A and C any two angles of a 

 triangle ABC, to meet each other at D a point in AC, and 

 the remaining lines AB, BC, in E and F ; their other in- 

 terfecflion H, the remaining angle B, and the points E, F, 

 are in the circumference of a ciixle. 



Join DH, EH, FH. The angle AEH is equal to ADH or 

 CFH, that is, BEH is equal to BFH, hence the points H,B, D, F 

 are in a circle. Q^ E. D. 



PROP. V. PORISM, Fig. 10. PI. II. 



Let AB, AC, BC be three ftraight Unes given by pofition, a 

 point H may be found, fuch, that if any circle be defcri- 

 bed through H, and B the interfedtion of any two of the 

 given lines, to meet them in D and F, and if DF be joined 

 meeting the remaining line at E. The line DF fhall be divi- 

 ded at E, into fegments having to each other a given ratio. 



Suppose that the point H is found. Join HA, HB, HC; join 

 alfo HD, HE, HF. Since, by hypothefis, a circle may pafs 

 through the point which is to be found, the interfedion of any 

 two of the given lines, and the points where DF meets thefe 

 lines, therefore the points H, A, D, E are in a circle, and the an- 

 gle HEF is equal to HAD or HAB ; now the points H, B, D, F 

 are fuppofed to be in ? circle ; fince therefore in the triangle 

 ABC, circles pafs through two of its angles A, B, and meet each 

 other at D, a point in AB, (Lemma.) the points H, C, E, F are 



O alfo 



