of the Trapezium. 

 Prop. VI. (Figs. 2 and 3.) 



119 



Upon the opposite sides ED, CK of a trapezium are con- 

 stituted the triangles EVD, CWK, each having its vertex in 

 the other's base. If S be the point of intersection of CW, 

 EV, and T that of VD, WK; then will ST always pass 

 through L, the intersection of the diagonals CD, KE of the 

 trapezium*. 



Dem. — Let A be the intersection of K C, ED, and ST cut 



CK, ED in I and F. Then 



CS . SE : VS . SW : : CA . AE : AV . AW, 



VT . TW: KT . TD : : AV . AW : KA . AD (ib.); hence 



CA.AE:KA.AD 



CL . LE : KL . LD (a) 



: SC . SE : VS . SW {ib.) 



: VT . T W : KT . TD (ib.) hence 



:IC.FE:IK.FD (b) 



VS.SW VT.TW 



Again, 



IC. FE : 

 IV. FW 



CS.SE 



IV . FW: 

 IK. FD: 



KT.TD 



VS. SW VT.TW 



* This theorem also includes a considerable number of 

 however it is unnecessary to insert here. 



jures, which 

 Hence 



