some Properties of Tangents to Circles; 3^^ 
And LGI (LGS and IGS) is equal to LSG and ISG together, 
that is, to LSI, which, Iiaving the supplement LGH, is equal 
to LGY. g. £. D. 
Cor. 1. The angle HIT is bisected by LR, which bisects HT 
perpendicularly : and GIH is bisected by NO, because HIR is 
equal to TIR, so to GIL. 
Cor. 2. In every triangle in which IK is one side, and the 
opposite angle is at the circumference of the circle, the two 
other sides, adjacent to IR and RK respectively, are * in the 
ratio of IR to RK. 
Cor. 3. Fig. 6 . If a right line IK be divided unequally in R, 
in a ratio given ; a circle LNR may be found such, that, IK 
being a side of any triangle having the opposite angle at the 
circumference, one of the other sides shall bear to the remaining 
side the given ratio. Describe the semicircle ITK, draw RT 
perpendicular to IK, and the tangent TL cutting KI produced 
in L: a circle LNR, on the diameter LR, is the circle required. 
For, let the two circumferences cut each other in Q; bisect 
IK in F, LR in C ; draw IN perpendicular to LR ; also FQ, 
CQ, FT, CN, KN. Then, since LTF and TRF are right 
angles, the rectangle LFR is -f equal to the square of FT, so 
of FQ; whence FQ touches J LNR, FQC is a right angle, and 
CQ touches § ITK; whence the rectangle KCI is || equal to 
the square of CQ, so of CN, and the angle KNC, as equal ^ 
to NIC, is a right angle, therefore KN a tangent. 
Prop. VI. Fig. 5. If the tangent-chord of a given point be 
* VI Eucl. prop. 3. f VI Eucl. prop. 8. Cor. and prop. 17. 
t III Eucl. prop. 37. § III Eucl. prop. 16. Cor. |j III Eucl. prop. 36. 
^ VI Eucl. prop. 14 and 6. 
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