some Properties of Tangents to Circles; 357 
And LM is * equal to LG, and RM to RS; whence the 
squares of LM and RM are together equal to that of LR. 
Therefore LMR is a right angle. Q. E. D. 
Cor. 1. If V be the vertex to LR, and right lines from L 
and R to E, also to V, be drawn or conceived; the angle LER 
is acute, LVR obtuse.]; 
Cor. 2. If the tangent-chord of a point L will pass through 
a point R, and if D in LR make RLD equal to the square of 
LG ; ED is perpendicular to LR. For, draw EH. Then RL 
is to LG as LG to LD ; whence the angle LDG is § equal to 
LGR or LGH, so to LHG, and LGEHD is one circle. But 
LHE is a right angle ; therefore also LDE. 
Prop. IV. Fig. 4. If two points be such, that, the intercepted 
right line being a base and the mean point to it being found, 
lines from this to the given points contain a right angle, or 
that the squares of the tangents from the two points are to- 
gether equal to that of the intercepted line; the tangent-chord 
of each of the two points will pass through the-other. 
Thus : if, LR being a base, LMR at the mean point be a 
right angle, or the squares of LG and RS be together equal to 
that of LR ; the tangent-chord of L will pass through R, and 
that of R through L. 
Dem. On the first supposition, the tangent- chord of R will 
pass through L : else it would cut the indefinite base in some 
other point, from which a right line to M would be || perpen- 
dicular to MR; which is impossible. Therefore also ^ the 
tangent- chord of L will p^s through R. 
On the second supposition, LMR will be** a right angle; 
* Prop. II. t I Eucl. prop. 48. J I Eucl. prop. 21. 
§ VI Eucl. prop. 6. || Prop. III. ^ Ib. ** Dem^ of Prop. HI. 
MDCCCXIV. 3 A 
