some Properties of Tangents to Circles ; ^c, 365 
Cor. 1. ABC is always obtuse, and AFC acute; because 
their sum is two right angles \ whence AC is not a diameter. 
Cor. 2. BF, subtending the mean angles, is greater than the 
other diagonal. For, if they be right angles, BF is a diameter. 
If they be not, one is acute ; as BAF. Then, since this is 
greater than AFC, the double of it, that is, an angle on BF at 
the * centre, is greater than one on AC at the centre ; there- 
fore -f BF than AC. 
Cor. 3. The greater diagonal, produced, cuts LR between 
L and R ; the less beyond L or R, if at all. 
Cor. 4. The sides containing the remote angle, are greater 
than their opposites; FC than AB, AF than BC. For the 
triangle LAC is similar to LBF, and LCF to LAB. Therefore 
LF is to LC as BF to AC; whence LF isj greater than LC, 
and than LB. But LF is to LB as FC to AB. By the like 
proof, AF rs greater than BC. 
Prop. IX. Fig. 9. The tangent-chord of each point of con- 
course, of an inscribed trapezium, will, produced, pass through 
the other. 
Thus : if L and R be the points of concourse ; the tangent- 
chord of L will pass through R, and that of R through L. 
Dem, Let ABCF be the trapezium, ABC the near angle, 
AFC the remote, and L the concourse of FA and CB. 
Describe the circle ABL, meeting LR in D; and draw AD, 
also the tangents LG, RS. Then the angle LDA is equal to 
LBA, which, by the common supplement ABC, is equal to 
LFC or LFR; whence the triangle LDA is similar to LFR, 
and the rectangle RLD equal to FLA, so to the square of LG. 
But LRD IS equal to ARB, so to the square of RS. Therefore 
I Eucl. prop. 24. J Cor. 2. 
3 
* III Eucl. prop. 20. 
MDCCCXIV. 
