of the Trapezium. 



117 



ECKD a trapezium, — whose opposite sides meet in A and 

 B and whose diagonals intersect each other in L, — is cut by 

 the transversa] I, F as in the figure. 



Fig. 1 



Prop. I. 

 FD . CI : FE . KI : : CG . DH : GE . KH * 



Dm.— Draw ME parallel to KD, and KN to CE. Then 



EG : EM : : KN : KH, 



KN:CG ::IK :IC, 



EM: DH : : EF : DF; and compounding 



FD.CI:FE.KI::CG.DH:GE.KH. q.e.d. 

 Cor. 1. When F and I coincide with A, we have 

 DA . AC : EA . EA . AK : : CG . DH : GE . KH, and 



.*. CG . DH : GE . KH a constant ratio when the transversal 

 passes through A. 



Cor. 2. WhenCG.DH:GE.KH::DA.AC:GE.KH 



the transversal passes through A. 



Cor. 3. When the transversal passes through A and B 

 CA . AD : KA . AE : : CB . BD : EB . BK. 



Prop. II. (Fig. 1.) 



IC . FE : IK . FD : : EO . PC : OK . PD. 



Bern.— Draw ME parallel to KD, and KN to CE. Then 

 FE : FD : : EM : DP, 

 EM:KN::OE:OK, 



I C : IK:: PC: KN ; hence, compounding 

 IC.FE:IK.FD::EO.PC:OK.PD. o..e.d. 



* To meet all the varieties which result from taking the trapezium suc- 

 cessively salient, re-entrant, and inter sectant, and from interchanging CandK 

 in each of these classes, would require upwards of forty figures ; but it is 

 unnecessary to insert them here, as the geometrical reader will easily sketch 

 them for his own use. Forty-two is the number I have drawn, but I can- 

 not positively affirm that all possible cases are included in this list. 



Cor. 



