some Properties of Tangents to Circles; &c. 361 
is parallel to KC: and CZ is to ID as GC to GI, as * TC to 
TI, as CZ to IW. And, conversely, if ID be equal to IW, 
then GD and WT, produced, will meet in a point Z of the 
base. 
Cor. 3. Fig. 6. If L, I, R, K, in that order, be points in one 
right line, and LK be to RK as LI to IR, and if LR be a dia- 
meter of a circle LNR, and IN perpendicular to LR ; KN is a 
tangent. For, if not, the tangent- chord of K will cut LR in 
some point X other than I. Then LX is to XR as LK to RK, 
as LI to IR: which is impossible. So, if IK be the diameter of 
ITK, and RT a perpendicular ; LT is a tangent. 
Cor. 4. Fig. 6. If any three of L, I, R, K, be given; the 
fourth, if an extreme point, as K, may be found by LNR, IN, 
NK; if an intermediate point, as R, then by ITK, LT, TR. 
Cor. 5. Fig. 7. A circle PHF may be described, having a 
given vertex V to a given base LR, and passing through a 
given point P whose perpendicular distance from LR is greater 
than from VF parallel to LR. Let GD be a perpendicular, 
through V, on LR. Draw PD, PV ; also PG, making the 
angle VPG equal to PDG. G is the centre required. For, 
draw GF. Then, by the similar triangles DGP and PGV, 
the rectangle DGV is equal to the square of PG or GF ; 
whence the triangles DGF and FGV are similar, DFG is a 
right angle, as equal to FVG, and DF a tangent; whence V 
is vertex in PHF to LR. Let DV cut the circle first in H, 
then in K. DK is to KV as DH to HV. Therefore DH is 
greater than HV ; and, if P were not further from LR than 
from VF, the thing could not be done. If P be given in the 
* Cor. 1. 
f Cor. 1 to prop. I. 
