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 = AD^2, 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._

That a circle may always be drawn through three points, provided that they do not lie in a straight line, is proved only later on in Book IV .

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