368 Dr, R. Hey's Propositions containing 
is equal to the square of* half of the tangent-chord of D 
therefore to AVC, and to BVF. 
Cor. 3. Fig. 10. If ABCF be an inscribed trapezium, having 
V the mutual intersection of the diagonals, L and R the points 
of concourse, ABC the near angle, AFC the remote, and A in 
LF, and if FB and AC, produced, cut LR in I and K; LK is 
to RK as LI to IR. For, let LB and RB, produced, cut SP, 
drawn through V parallel to LR, in P and S. Then LK is to 
PV as CK to CV, as-f AK to AV, as RK to SV ; whence LK 
is to RK as PV to SV, as LI to IR. 
Cor. 4. Fig. 10. If to M, the mean point to the base LR, 
be drawn LM, IM, RM, and KM produced to any point X ; 
the angle IMK is bisected by MR, and IMX by ML. For, on 
the diameter LR, and on the same side with the trapezium, 
describe the semicircle LNR, and draw IN perpendicular to 
LR. Then KN, being drawn, is;j] a tangent. And, because 
LMR is§ a right angle, M is in the circumference LNR; 
whence II the angles are bisected. 
Cor. 5. Fig. 10. Or, if a base LK be cut in any two points 
I and K by chords, as FB and AC, through its vertex V, and 
M be the mean point ; the angle IMK is bisected by drawing 
either AB or FC, producing it to R in the base, and drawing 
MR. For AB and FC meet the base in one point: and the 
inscribed trapezium ABCF may be completed. Also IMX 
might be bisected in a similar manner. 
Cor. 6. Fig. 9. If any polygon, having an even number of 
sides, be inscribed in a circle, and its diagonals joining opposite^ 
* Of PV in Fig. 2. f Cor. 1 to prop. VI. J Cor. 3 to prop. VI. 
§ Cor. I to prop. IX. || Prop. V. 
^ By opposite angles and sides are meant such, that the two intercepted arcs contain 
equal numbers of angular points. 
