§ 37. If we draw chords through a point A within a circle, they will each be divided by A into two segments. Between these segments the law holds that the rectangle contained by them has the same area on whatever chord through A the segments are taken. The value of this rectangle changes, of course, with the position of A.
A similar theorem holds if the point A be taken without the circle. On every straight line through A, which cuts the circle in two points B and C, we have two segments AB and AC, and the rectangles contained by them are again equal to one another, and equal to the square on a tangent drawn from A to the circle.
The first of these theorems gives Prop. 35, and the second Prop. 36, with its corollary, whilst Prop. 37, the last of Book III., gives the converse to Prop. 36. The first two theorems may be combined in one:—
If through a point A in the plane of a circle a straight line be drawn cutting the circle in B and C, then the rectangle AB.AC has a constant value so long as the point A be fixed; and if from A a tangent AD can be drawn to the circle, touching at D, then the above rectangle equals the square on AD.
Prop. 37 may be stated thus:—
If from a point A without a circle a line be drawn cutting the circle in B and C, and another line to a point D on the circle, and AB.AC = AD², then the line AD touches the circle at D.
It is not difficult to prove also the converse to the general proposition as above stated. This proposition and its converse may be expressed as follows:—
If four points ABCD be taken on the circumference of a circle, and if the lines AB, CD, produced if necessary, meet at E, then
EA·EB = EC·ED;
and conversely, if this relation holds then the four points lie on a circle, that is, the circle drawn through three of them passes through the fourth.