some Properties of Tangents to Circles; &c. 355 
And the vertex to a given base LR, if E be the centre and 
ED a perpendicular on LR, is the intersection of ED with the 
tangent-chord of any point D or L of the base. 
Cor. 2. Fig. 1. The mean point M, to a given base, is in the 
same diameter with V : and DM, as equal to DP, is a mean 
proportional between DE and DV. For, if EP be drawn, the 
triangles EDP, PDV, are similar. 
Cor. 3. Fig. 2. If a chord GH bisect PQ the tangent-chord 
of any point D ; DE bisects the angle GDH. For, draw DG, 
DH, EG. The right lines EG, EH, are equal; and the arcs 
EG, EH ; whence the angle GDE is* equal to EDH. And, 
conversely, if DE bisect GDH, GH bisects PO the tangent- 
chord of D. For, if any other chord through G should bisect 
PQ. the same angle GDE would be equal to one greater or 
less than EDH : which is impossible. 
Cor. 4. Fig. 2. V is the vertex to the base LR in every circle 
whose centre, as E, is in DV produced beyond V, and whose 
radius, as EP or EO, is found by the intersection P or Q of a 
circle on the diameter ED with the parallel PQ. For DPE 
and DQE are-f right angles ; whence DP and DQ are J 
tangents. 
Prop. II. Fig. 3. The distance of an assumed point, in a given 
base, from the mean point within the circle, is equal to a tangent 
from the assumed point.§ ' 
Thus: if L be assumed in the base LD, and if M be the 
mean point within the circle GBQ, and LG a tangent; LM is 
equal to LG. 
* III Eucl, prop. 27. f III. Eucl. prop. 31. J III. Eucl. prop. 16. Cor. 
^ This is a lemma in Hamilton’s Stereography, with a different proof. B. HI. 
Sect. I. L. 18. 
