35^ R. Hey’s Propositions containing 
Dem. From the centre E draw ED perpendicular to LD; 
and draw the tangent DQ ; also EQ, EG, LE. The squares 
of LM and radius are equal to those of LD and DM (or DQ) 
and radius ; therefore to those of LD and DE, so to that of LE 
or those of LG and radius ; whence the square of LM is equal 
to that of LG, and LM to LG. Q. E, D. 
Prop. III. Fig. 4. If the tangent-chord of any point will, 
produced, pass through any other given point ; that of the 
second will pass through the first ; and the squares of the two 
tangents, one from each point, are together equal to that of 
the intercepted right line ; and, such line being taken as a base, 
and the mean point to it being found, lines to this from the two 
points contain a right angle. 
Thus : if ST, the tangent-chord of R, will pass through L ; 
GH, that of L, will pass through R ; and the squares of the 
tangents LG and RS are together equal to that of LR; and, 
LR being a base, LMR at the mean point is a right angle. 
Dem, Draw LH ; and let it be between LR and LG. Draw 
from the centre E, to LR, a perpendicular ED ; also GD, GR. 
Describe the circles LGEH, RSET *. These will pass through 
D, because of the right angles LDE, RDE ; whence the 
rectangle RLD is equal to SLT, so to the square of LG, and 
LD is to LG as LG to LR. Therefore the angle LGR is-f* 
equal to LDG, to LHG, to LGH ; and GH produced will pass 
through R. 
And, because of the circle GHD, the rectangle LRD is 
equal to GRH, so to the square of RS ; whence this square and 
that of LG (proved equal to RLD) arej together equal to that 
ofLR. 
* As LGEH in Prop. I. t VI. Eucl. prop. 6. j II. Eucl. prop. 2 . 
