And the converse (which is the more useful part of the Theorem) follows easily by an indirect proof, similar to that of Theorem (4).

THEOREM (6)—If two chords of a circle intersect within it, the angle between them is equal to the circumferential angle on an arc which is equal to the sum of the arcs subtended by that angle between the chords which is under consideration.

THEOREM (7)—If two produced chords of a circle intersect without it, the angle between them is equal to the circumferential angle on an arc which is equal to the difference of the arcs intercepted between the chords.

Let AB, CD be chds. of a O, which, being produced, meet at pt. O, outside it. Draw chd. DX || to AB.

Def. A sector of a circle is the plane figure contained by two radii and the are they intercept.

THEOREM (8)—In equal circles (or the same circle) sectors on equal arcs are identically equal.

THEOREM (9)-[Converse of iii. 35 and Cor. (a) to 36.] If four points are so situated that the rectangle under the distances of two of them from the intersection of their joins (or joins produced) is equal to the rectangle under the distances of the other two from the same intersection, then the four points are concyclic.

P B

Let A, B, C, D, be four points such that joins AC, BD meet in X; and joins BC, AD produced, meet in Y.

r, let XA . XC Assume that

through three of the pts, say C, D, A, meets BD in some

But this is absurd, for one of them is a part of the other.

through C, D, A meets CB in some pt. P (not B).

And again this is absurd, for one of them is a part of the other.

.. in both cases the assumption that A, B, C, D are not concyclic leads to an absurdity; and .. is not true :

i. e. A, B, C, D are concyclic.

SOME USEFUL THEOREMS, MAINLY DEPENDING ON BOOK iii.

THEOREM (10)—If each side of a quadrilateral touches the same circle, the sum of one pair of its opposite sides is equal to the sum of the other pair; and conversely.

O with O as centre, and any one of them as radius, will go through P, Q, S; and will touch AB, BC, DA at those pts., the As at P, Q, S are right.

Assume that this O does not touch CD.

Draw CX a tangent to the O, so that it meets AD (or AD produced) in X.

.. the assumption that O does not touch CD leads to a contradiction; and .. is not true:

i. e. sides of quad. all touch same O.

S

=

Note-From preceding, and iii. 22, we see that if the opposite As of a quad. the sum are supplementary, and also the sum of one pair of opposite sides of the other pair, the quad has its corners on one circle, and its sides touch another or (see Defs. of iv) it circumscribes one O, and is inscribed in another.

THEOREM (II)—The bisectors of the angles formed by producing the opposite sides of a cyclic quadrilateral to meet, are at right angles.

THEOREM (12)-(Brahmegupta's) If the diagonals of a cyclic quadrilateral are at right angles, the perpendicular from their intersection on any side, being produced, bisects the opposite side.