Lawson's Geometrical Theorems. 425 
+ 2F3? = 2q? = 2FD. FH + 2F3* = 
2H3.sh. Hence if from any point F in 
the diam. of a circle produced, a perpen- 
dicular be raised, and thereon F’d be taken 
whose square is equal to the rectangle 
DFH, and Fa be put equal to F’d and 
from any point in ’d.a as 3a line be drawn 
to cut the circle in any two points as:h 
and Hi, then twice the rectangle h3H is 
equal to the sum of the squares of sd 12 
and ja. «Which is the twelfth Proposition. 
If 3s be drawn through q cutting the 
circle in r and s, themthe rectangle =:res 
= hoyH = (by the last) *q*. Which is the 13 
thirteenth Proposition. | 
If in AB the diameter of a circle produced. a point C be 
taken, and therefrom a tangent as CD be drawn, and in the 
diameter a point E be taken such that AC: CB: : AE : 
EB ; then I say ED being drawn will be perpendicular to 
‘the diameter AB. 
Prop. XVI. Let AB be any chord in a circle and CD 
another cutting the former in E, CB being jomed, from D 
draw DF parallel to CB to meet AB in F; I say that the 
rectangle AEF is equal to the square of DE. 
Prop. XVII. It ABC be a iriangle inscribed in a circle 
whose sides CA and CB are equal, and the rectangle CBD 
Equal to the square of AB, and let AE be any line cutting 
CB in Fand the circle again in E, and from E let a parallel 
to AB be drawi to meet CB in G; then I say that the 
rectangle OFG : BF?-::. CG: BD. ’ 
3H 
