Lawson’s Geometrical Theorems 423 
Diacram II. 
Perpendicular to n the center, or any 
other point of the Diameter DH, erect 
ng, and make the » gf'n — FEG; then 
Fon = GEe ; produce F¢ till it meets He 
in m; then ¢F'n = heh’ — hnF'; therefore 
Fm is parallel to he, and lG: GE:: Fh: 
FE ::em: Em. Which is Proposition 10 
tenth, part 1st. 
Also if through the point h, any line 
hL be drawn to the circle, and at G the 
zg¢GM be made equal to LhE, the points 
M, G, g, hare ina circle, therefore the 
rectangle FHG is equal to the sum of the squares of HD 
and HE. 
Prop. XIII. If in AB the diameter of a circle two 
points C and D be so taken that, C being without, and D 
either within or without the circle, the square of CD be 
equal to the rectangle ACB, and from C a perpendicular 
to AB erected, and any line drawn through D to cut the 
same in G and the circle in E and F ; then I say the square 
of GD will be equal to the rectangle EGF. 
The converse is also true, which is this. 
If GC he perpendicular to AB the diameter of a circle 
and meets it without the circle in C, and if from Ga line 
be drawn to cut the circle in E and F, and the diameter 
either within or without in D, and the square of GD be 
