Lawson's Geometrical Theorems 427 
Join TH, T’H and from T’ set off T’7B 
so that T'l’* = the rectangle H'T’B, and 
let TE be any line cutting T’H in A, and 
the circle again EK’ and from HE’ let a pa- 
rallel to T'T’ be drawn to meet T’H in », 
then (by the ast) AE’* = Ay. AH and 
the As AvE’, AHE’ and consequently 
AT’T similar, as also K’H, THE’, there- 
fore as AK’? — Ay. AH: AT’ : : E’'H? 
: TT’; butyH : KH::E’H: TH, .:. F'H? 
=—7H.TH: TT? = T'B.TH:: 7H: 
T’B; consequently HA. Ay: AT’? : : YH: 17 
TB. Which is the seventeenth Proposition. 
Let Hh cut TT’ ini, then GH: TH: : 
TH : DH and GH : iH::hH: DH; 
Prop. XXI. If in AB the diameter of a circle be taken 
two points C and D such that AC:CB:: AD: DB, and 
D be within the circle, and DE be perpendicular to AB 
meeting the circle in E and F, and if through C any line 
be drawn meeting the circle in G and H, andthe line DE 
in K, and GL touch the circle in G, and meet DE in L; 
then I say the rectangle LDK is equal to the “na 
of DE. 
Prop. XXII. If in AB the diameter of a circle be 
taken two points C and Dsuch that AC: CB:: AD: DB, 
and D be without the circle, and DE be perpendicular to 
AB, and through C be drawn any line meeting the circle 
in G and H, and the line DE in K, and GL touch the 
circle in G, and meet DE in L; then I say the rectangle 
LDK is equal to the rectangle ADB, 
oH 2 
