202 CEOMETRICAL THEOREM 



is to BE so is BO to AB and so is FN to CE; therefore as BO 

 is- to AB, so is FN to CE; and the rectangle under BO, CE is 

 equal to the rectangle under AB, FN. But the rectangle un- 

 der BD, CM is also equal to the rectangle under AB, FN, it 

 is therefore equal to the rectangle under BO, CE. Therefore 

 as BD is to CE, so is BO to CM. Through H draw HI, HP, 

 parallel to AB and AC. Then hecause EH is equal to HD, 

 and HI is parallel to BD, BE is bisected in I, and HI is one 

 half of BD. In the same manner CD is bisected in P, and 

 PH is one half of CE. Bisect BC in K and draw KP, and 

 KI which produce to S and T. Then because CK is equal to 

 KB, and CP is equal to PD, KP.is parallel to BD and equal to 

 one half of BD, and in the same manner KI is parallel to CE 

 and equal to one half of CE; and K, P, H, I is a parallelogram. 

 And CS is equal to AS, and BT to AT. Through G draw VG 

 parallel to AC, and produce VG to X, cutting CD in X, KS 

 in W, and HI produced in Z : draw XY parallel to AB. 1 hen 

 because AG is equal to GF and VG is parallel to AC, and con- 

 sequently to OF, AV is equal to VO; But AT is equal to BT, 

 therefore BO which is equal to the difference between twice 

 AT and twice AV, is equal to twice TV. Because AG is 

 equal to GF and GX is parallel to AC, FX is equal to CX, 

 and because XY is parallel to AB and consequently to FM, 

 CY is one half of CM; but CS is equal to SA. And AM 

 which is equal to the difference between twice CS and twice CY 

 is equal to twice SY. Because GA is equal to FG and GX is 

 parallel to AC, GX is equal to one half of AC, i't is therefore 

 equal to CS. WX is parallel to SY, and SW to XY, there- 

 fore SWXY is a parallelogram and SY is equal to WX, GW 

 is therefore equal to CY, and CM is equal to twice GW; and 

 because KW is parallel to TV and VW to KT, KTVW is a 

 parallelogram and KW is equal to TV, and BO is equal to twice 

 KW. But as BD is to CE so is BO to CM, that is as twice KP 

 is to twice PH so is twice KW to twice GW, so as KP is to PH 

 so is KW to GW, and therefore as KP is to the difference be- 

 tween KW and KP, so is WZ which is equal to PH to the dif- 

 ference between GW and WZ, that is as KP is to HZ which is 

 equal to PW so is PH to ZG. Join GH and HK; now the tri- 



