some Properties of Tangents to Circles; ^c, 3^^ 
angles meet in one point V, not the centre ; the opposite sides 
will respectively meet, if at all, in points of the base to the 
vertex V. For any two opposite sides, as AB and FC, may 
be made the opposite sides of an inscribed trapezium having 
the same intersection of its diagonals. If two opposite sides 
be parallel, the base is* parallel to them. 
Cor. 7. Fig. 11. If ABCF and LR be as in the proposition, 
and if IK, KN, NO, OI, be tangents respectively at A, B, C, 
F ; the diagonals IN, KO, of the circumscribed figure IKNO, 
cut each other in the same point V as do those of ABCF, and 
are in the same indefinite lines as the tangent-chords of L and 
R. For, since AB the tangent-chord of K will pass through 
R, that of R wilFf- pass through K ; likewise through O, whose 
tangent-chord is FC. And it will pass through J V, and§ L. 
In like manner the tangent-chord of L will pass through N, 
I, V, and R. 
Cor. 8. Fig. 11. Therefore the opposite sides of a circum- 
scribed trapezium, if they meet, meet in points of the same' 
indefinite base as that of such inscribed one; namely, whose 
angular points are the points of contact of the circumscribed. 
For the sides are tangents from chords through V. If two 
opposite sides be parallel, the base is parallel to them. For, 
if 10 and KN, which are tangents at F and B, were parallel, 
FB would be a diameter, and perpendicular to them: and, 
because it passes through V, it would be the diameter perpen- 
dicular to LR. 
Cor. 9. Fig. 11. Also, if a circumscribed polygon have an 
even number of sides, and its diagonals || pass through one 
* Cor. 1. 
§ Prop. IX. 
f Prop. III. 
II As in Cor. 6. 
I; Prop. I. 
