118 Mr. T. S. Davies's Properties 



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

 CA . AE : KA . AD : : CP . EO : OK . PD. 



Cor. 2. When O and P coincide with L, we have 

 IC . FE : IK . FD : : CL . LE : KL . LD. 



Cor. 3. When the transversal passes through A and L, 

 we find that 



• CA . AE : KA . AD : : CL . LE : KL . LD. 



Cor. 4. When F and I coincide with A, we have 

 CL . LE : KL . LD : : CP . EO : OK . PD. 



Cor. 5. Similarly we find when O, P, L coincide 

 IC . FE : IK . FD : : CA . AE : KA . AD. 



Prop. III. (Fig. 1.) 



GC.HK:GE.HD::KO.CP:OE.PD. 

 Bern.— Draw EM, KN as before, and KR parallel to CD. 

 Then CG : KN : : CP : KR, 



KN : GE : : KO : OE, 

 KH : K 3 R : : H 2 D : PD ; hence 

 CG . KH : GE . HD : : KO . CP : OE . PD. g. e. d. 



Cor. 1. When the transversal passes through L, we find 

 CG . KH : GE . HD : : CL . LK : EL . LD. 

 ::CB.BK:EB.BD. 



Prop. IV. (Fig. 1.) 

 Let the points ECKDlie in the circumference of a circle; then 



AE : AK : : CL : LK : : EL : LD : : AC : CD. 

 Dm.— By circle AE . AD : AC . AK : : CL . LD : EL . LK, 

 and (Cor. 1. Pr. 2.) AE. AC: AD. AK: : CL. LE: KL. LD; 

 hence, &c. q. e. d. 



Prop. V. (Fig. 1.) 

 The points ECKD being still in the circle we shall have 

 KC . CE : KD . DE : : CL : LD, and 

 KC.KD:CE. ED::KL:LE. 



Devi. — The triangles CLE, KLD are similar, and 

 CE : KD : : EL : LD 



CK : ED : : CL : LE ; hence compounding 

 KC.CE:KD.DE::CL:LD. 



In like manner is the second analogy demonstrated, o. e. d. 



Cor. Hence also CE 2 : KD 2 : : CL. LE : KL. LD, 



: : CA . AE : KA . AD, 

 and CK 2 : ED 2 : : CL . LK : EL . LD 



::BC.BK:EB.BD. 



Prop. 



