Lawson's Geometrical Theorems.. 431 
Eh’, F’K, Gh’e’ are similar as also Ghg, . 
and the points G, h, 4, E are in a circle, 
the triangle ,agE, and consequently LIE is 
similar to hgG, and consequently to Fh’K, 
therefore FK : Fh’: : FE : Fl and FK . 
Fl= Fh .FE=FD.FH. Which is the 
twenty-second Proposition. 
Produce hD to y, then by similar trian- 
gles FD: Fy: : Fo : FH. Which is the 23 
twenty-third Proposition. . 
| From hand E to any point N in the 
_circle, let lines be inflected cutting G in 
k and L, then because the angle ENh = 
EE‘ = LGh the points h, L, N, G are 
ina circle, consequently Lk .kG@ =kh . 
22 
the diameter AB in L; then I say that AL: LB:: | 
AC : CB. 
Prop. XXIX. Let AB touch a circle in B, and any 
line AE be drawn equal to AB, and likewise from A let 
any line be drawn to cut the circle in C and D, and let 
EC, ED be drawn meeting the circle again in F and G; 
then FG being drawn will be parallel to AE. 
Prop. XXX. Let AB touch a circle in B, and 
therein be taken two points E and F on the same side of 
A such that the rectangle EAF may be equal to the square 
of AB, and from A let any line be drawn meeting the 
circle in C and D, and EC, FD be drawn meeting the 
circle again in G and H; then GH being drawn will be 
parallel to AB, 
Prop. XXXI. Let AB touch a circle ia B, and any 
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