GEOMETRICAL PORISMS. 125 



Construction. Through A, B, C defcribe a circle ; inflecfl 

 BH, CH to the circumference, fo that BH may be to CH in the 

 given ratio of BD to CE, or of M to N*, thus H will be a given 

 point. If the fegments BD, CE to be cut off, are to lie in the 

 fame diredtion with AB, AC, the point H mufl be found in the 

 fame fegment with BAG ; but if they are to lie ii\ contrary di- 

 re<5lions, then H muft be taken in that fegment upon which 

 BAG flands. Join AH and PH, iipon which defcribe a feg- 

 ment of a circle, that may contain an angle equal to HAC, 

 which is given. This circle may cut AB in two points D, 5. Join 

 PD and Fi, meeting the remaining line in E and s ; thefe lines 

 cut off fegments BD, CE, or BJ, Cs, having to each other the 

 given ratio of BH to HC, or of M to N. 



Join HD, HE. Becaufe the angle PDH is by conftrudlion 

 equal to HAE, the points A, H, D, E are in a circle ; therefore 

 the angle HEA is equal to HDA, that is HDB is equal to HEC; 

 now, HBD is equal to HCE, for HBA is equal to HCA, there- 

 fore the triangles HCE, HBD are fimilar, and BD is to CE as 

 BH to HC, that is, by conflrudion, as M to N. 



It is evident that this problem may admit of four folutions in 

 general, if there be given no limitation with refpedl to the di- 

 rection in which the fegments are to be cut off from the given 

 lines ; but the data may be fuch as to render it capable of three 

 and alfo of two folutions only. 



The next example Ihall be the Se6lio fpatii of the ancients. 



PROP. XII. PROBLEM, Fig. 17. PI. m. 



Two flralght lines AB, AC are given by policion, and two 

 points B, C are given in thefe lines. It is required to draw 



a 



» The manner of doing this has been fhewn in Prop. t. 



