J 24> Mr. T. S. Davies's Properties of the Trapezium. 



BK . BI : BL . BH : : AF . CK . CG . AK : AL . CF . CH 

 GA. 



Fig. 10. 



<* p 



2. Again, the triangles ALI, CKH, cut by the tranversal 

 FG in the points F and P", give 



LF : FI : : GA . FL : IG . AF (Bland, ib.) and 

 HP": P"K : : FC . GH : KF . CG. Whence, comp. 



LF . P"H : FI . F'K : : G A . FL . FC . GH : IG . AF . KF . 



CG. 



3. Further, the triangles AFC, ACG give 



LF : LA : : BC . KF: AB . CK. (Bl. ib.) and 

 GH : CH : : GI . AB : BC . IA. (ib.) Hence 

 LF. GH _ IG. KF 



CH. AL ~" CK. AI * 



4. Dividing the terms of the first proportion by the corre- 

 sponding terms of the second, we have 



LP'. HP" FI.P"K FL.GH IG.KF „ x , ,_; 

 LB.HB : ^rBK- :: -cHTAL : CK7Ai- But by (3) we 



learn that the second of these is a ratio of equality ; hence 

 LP'. HP" _ PI . F'K 

 LB.HB ~~ BI.BK' 01 

 LF . HP" : FI . F'K : : LB . BH : KB . BI, 



: : LP . PH : IP . PK (Cor. 3. Pr. 2), 

 and hence (converse of Cor. 2. Pr. 3.) the lines LI, KH, and 

 FG intersect in the same point P. g. e. d. 



Cor. 1. Conversely we find, ex absurdo, that when LI, 

 KH, situated as in the theorem, intersect in FG, the lines LK, 

 HI, will intersect in AC. 



Schol. The three figures above given are sufficient for the 



purpose 



