GEOMETRICAL PORISMS. 125 



Construction. Through A, B, C defcribe a chxle ; infle(5l 

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

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

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

 fame diredlion Wixh AB, AC, the point H muft be found in the 

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

 rections, then H mufl be taken in that fegment upon which 

 BAC flands. Join AH and PH, upon which defcribe a feg- 

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

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

 PD and P^, meeting the remaining line in E and £ ; thefe lines 

 cut off fegments BD, GE, or B^, Cg, 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 conflrudion 

 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 HGA, there- 

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

 BH to HC, that is, by conftrudion, 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- 

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

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

 and alfo of two folutions only. 



The next example (hall be the SeBlo Jpatii of the ancients. 



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



Two ftraight lines AB, AC are given by pofition, and two 

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



a. 



* The manner cf doing this has been {hewn in Prop. i. 



