370 Dr. R. Key's Propositions containing 
point V, not the centre; the opposite sides, if they meet, meet 
respectively in * one base ; and it is tlie base in which those 
of such 'f inscribed one meet. If two opposite sides be paral- 
lel, the base is parallel to them. 
Prop. XI. Fig. 13 . If, in each of two circles having a com- 
mon vertex to a given base, a chord be drawn through the 
vertex, and if, by joining the extremities of the two chords, a 
trapezium be formed having two points of concourse; the base 
is the connecting line. If two sides of the trapezium be paral- 
lel, they are also parallel to the base. 
Thus: if, in BFT and ACK, BF and AC be chords through 
V the common vertex to the base LR, and if ABCF have two 
points of concourse; these are in LR. If FA and CB were 
parallel, they would be parallel to LR. 
Dem. Through V draw NO parallel to LR. The opposite 
sides FA and CB, since they meet, cannot both be parallel to 
LR. Let FA, produced beyond A, meet LR in L, and cut NO 
m N. Through L and B draw LO, cutting NO in O ; and 
draw LC, OC. Then, by the circle BFT, NV and VO are^J] 
equal. And, because they are equal, then, by the circle ACK, 
AN (or FA) and OC produced will meet § in the base, there- 
fore in L; whence OC and CL are one right line OL; of 
which, LB is part. Therefore LBC is one right line, and L 
a point of concourse. In like manner, some point R of LR is 
the other point of concourse. Therefore LR is the connecting 
line. And since, if FA or CB meet LR, they are thus proved 
to meet each other, therefore, if they do not meet each other, 
neither of them meets LR; that is, they are parallel to it. 
Q. E. D. 
* Prop. J. t See Cor. 6. Cor. 2 to prop. VI. § Ib; converse. 
