Then, the exterior angle BDC of the triangle DEC is greater than the interior opposite angle DEC;

I. 9.

that is, the angle BDC is greater than the angle BEC which is in the segment ABC.

In like manner it may be shown that if D is a point outside the segment ABC and on the same side of the chord BC as the segment; then the angle BDC is less than the angle in the segment BAC.

COR. 2. The locus of a point on one side of a given finite straight line at which that line subtends a constant angle is an arc of which that line is the chord.

*Ex. 43. Two chords AB, CD of a circle intersect in E. Shew that whether E be within or without the circle the triangles AEC, DEB are equiangular. So also are the triangles EAD, ECB.

Ex. 44. BA is a chord of one, BC of the other, of two circles which intersect at B. Through B any straight line PBQ is drawn meeting the circumferences again in P and Q respectively. Shew that if PA, QC, produced if necessary, meet in R, R lies on a certain circular arc.

Ex. 45. P is any point on a circular arc APB, AP is. produced to Q, so that PQ is equal to PB: shew that the locus of Q is a circular arc.

THEOR. 17. The angle in a segment is greater than, equal to, or less than, a right angle, according as the segment is less than, equal to, or greater than, a semicircle.

Let BAC be an angle in the segment BAC of the circle ABEC:

then according as the segment BAC is less than, equal to, or greater than, a semicircle;

so shall the angle BAC be greater than, equal to, or less than, a right angle.

Let O be the centre.

Join OB, OC, if necessary.

Then, according as the segment BAC is less than, equal to, or greater than, a semicircle, the arc BEC is greater than, equal to, or less than, half the circumference of the circle, and therefore the angle BOC, which stands on the arc BEC, is greater than, equal to, or less than two right angles.

But the angle BAC at the circumference is half the angle BOC at the centre on the same arc BEC; III. 15.

therefore according as the segment BAC is less than, equal to, or greater than, a semicircle,

the angle BAC is greater than, equal to, or less than, a right angle.

Q.E.D.

Ex. 46. The circles described on two of the sides of a triangle as diameters intersect on the base, or the base produced.

Ex. 47. The four circles described on the sides of a rhombus as diameters have one point in common.

THEOR. 18. A segment is less than, equal to, or greater than, a semicircle, according as the angle in it is greater than, equal to, or less than a right angle.

Let BAC be a segment of the circle ABEC, BAC an angle in the segment:

then, according as the angle BAC is greater than, equal to, or less than a right angle:

so shall the segment BAC be less than, equal to, or greater than, a semicircle.

For in the preceding Theorem it has been shewn that

if the segment BAC is less than a semicircle,

then the angle BAC is greater than a right angle;

if the segment BAC is equal to a semicircle,

then the angle BAC is equal to a right angle;

and if the segment BAC is greater than a semicircle, then the angle BAC is less than a right angle.

Now of these hypotheses one must be true, and of the conclusions no two can be true at the same time;

hence, by the Rule of Conversion, the converse of each of the above Theorems is true.

Q.E.D.

DEF. 12. If the angular points of a rectilineal figure be on the circumference of a circle, the figure is said to be inscribed in the circle, and the circle to be circumscribed about the figure.

THEOR. 19.

The opposite angles of a quadrilateral

inscribed in a circle are supplementary.

Let ABCD be a quadrilateral inscribed in the circle ABCD :

then shall the opposite angles BAD, BCD be supplementary.

Let O be the centre of the circle.

Join OB, OD.

Then the angle BAD at the circumference standing on the arc BCD is half the angle BOD at the centre standing on the same arc BCD, III. 15.

and the angle BCD at the circumference standing on the arc BAD is half the angle BOD at the centre standing on the same arc BAD ;

therefore the sum of the angles BAD, BCD is half the sum of the two conjugate angles of which OB, OD are the arms; but the sum of two conjugate angles is four right angles, therefore the sum of the angles BAD, BCD is two right angles, that is, the angles BAD, BCD are supplementary.

Q.E.D.

COR. I. An exterior angle of a quadrilateral inscribed in a circle is equal to the interior opposite angle.

COR. 2. If the opposite angles of a quadrilateral are supplementary a circle can be described about the quadrilateral.

For if one of the angular points of the quadrilateral did not lie on the circumference of the circle (III. 12) passing through the other three, the angle at that point would not be supplementary to the opposite angle. III. 16, Cor. 1.

Ex. 48. Prove Theor. 19 by drawing the diagonals of the quadrilateral and shewing that the sum of the angles at the extremities of one diagonal is equal to the sum of the angles of either of the triangles into which the other diagonal divides the quadrilateral.

Ex. 49. If a circle can be described about a parallelogram, the parallelogram must be a rectangle.

Ex. 50. A circle can be described about every rectangle.

Ex. 51. The quadrilateral ABCD is inscribed in a circle; AB, DC produced meet in E, and BC, AD produced meet in F; prove that, if BEFD can also be inscribed in a circle, AC is a diameter of the first circle and EF of the other.

EXERCISES.

52. From the foot of a perpendicular AD on the hypotenuse of a right-angled triangle ABC, perpendiculars DE, DF are