g 66 Dr. R. Key's Propositions Gontaining 
the squares of LG and RS are together equal to RLD and 
LRD, so to the* square of LR; whence the tangent-chord of 
L willf pass through R, and that of R through L- g. E. D. 
Cor. 1 . If E be the centre, V the vertex to LR taken as a 
base, and M tiie mean point, and if rigiit lines be drawn or 
conceived, from L and R, to E, to V, to M; LMR is a right 
angle, LVR obtuse, LER acute.J 
Cor. 2. D is the dividing point. § 
Cor. 3. Each side of ABCF is a chord in a circle passing 
through the opposite point of concourse and the dividing point. 
For, describe the circle FCL; and let D be now the point in 
which this circumference cuts LR ; and draw DF. Then the 
angle LDF is equal to LCF, so to LAR; whence the triangle 
LDF is similar to LAR, and the rectangle RLD equal to FLA, 
and so to the square of LG, as before. Therefore D, as|| 
before, is the dividing point. And a circle CBR would be 
proved as ABL, and FAR as FCL, to pass through that point. 
Cor. 4. The diameter LI of ABL is perpendicular to FC. 
For, let them meet in K; and draw ID. Then the angle LID 
is equal to LAD, so to LRF or LRK; whence LKR, as equal 
to LDI, is a right angle. In like manner the diameter of FCL 
would be proved perpendicular to AB, that of CBR to AF, that 
of FAR to BC. 
Prop. X. Fig. 9. If an inscribed trapezium have two points 
of concourse; the connecting line is the base to which the 
mutual intersection of the diagonals is vertex in the inscribing 
circle. 
Thus: if ABCF have L and R its points of concourse; LR 
♦ II Eucl. prop. 2. f Prop. IV. j Prop. Ill, and its Cor. 1. 
§ Def. 5, and Cor. 2 to prop. III. (| This Prop, and Cor. 2. 
