364. 
Dr. R. Key’s Propositions containing 
Additional Definitions for the Introductory Propositions. 
Def. 4. Fig. 8. The points of concourse^ of a trapezium, are 
the two points in which (if at all) the opposite sides, produced, 
meet respectively. Thus : if ABCF be a trapezium, and if FA 
and CB meet in L, AB and FC in R ; the points of concourse 
are L and R. 
Def. 5. The connecting line is a right line joining the two points 
of concourse ; as LR. And, if the figure be one inscribed in 
a circle, LR is cut in the dividing point D by a perpendicular 
from the center. 
Def. 6. The near angle and the remote angle, of a trapezium 
having two points of concourse, are those, respectively, whose 
angular points have the least, and the greatest, perpendicular 
distances from the connecting line. The two others are the 
mean angles. 
Prop. VIII. Fig. 8. The near angle, of an inscribed trape- 
zium having tw’o points of concourse, is greater than any of 
the others; and the remote angle is less than any. 
Thus : if the figure be as in def. 4, and ABC be the near 
angle, AFC the remote ; ABC is greater than each of BAF, 
BCF, AFC ; which last is less than each of BAF, BCF. 
Dem. BAF, having * the supplement BCF, is equal to RCB ; 
and BCF to LAB. Therefore ABC, an exterior angle of the 
triangles RCB and LAB, is greater than either of the mean 
angles at A and C : and their equals, RCB and LAB, being 
exterior angles of the triangles LFC and RFA, are severally 
greater than AFC. g. E. D. 
