38. An isosceles triangle has its vertical angle equal to the exterior angle of an equilateral triangle. Prove that the radius of the circumscribing circle is equal to one of the equal sides of the given triangle.

39. Determine the centre of a given circle by means of a ruler with parallel edges whose breadth is less than the diameter of the circle.

40. From one of the points of intersection of two circles lines are drawn equally inclined to the common chord of the circles prove that the portions of these lines intercepted between the other points in which they meet the circumferences of the circles are equal.

SECTION IV.

ANGLES IN SEGMENTS.

DEF. 9. A segment of a circle is the figure contained by a chord and either of the arcs into which the chord divides the circumference. The segments are called major or minor segments according as the arcs that bound them are major or minor arcs.

DEF. 10.

The angle formed by any two chords drawn from a point on the circumference of a circle is called an angle at the circumference, and is said to stand upon the arc between its arms.

DEF. II. An angle contained by two straight lines drawn from a point in the arc of a segment to the extremities of the chord is called an angle in the segment.

THEOR. 15. An angle at the circumference is half the angle at the centre standing on the same arc.

Let BAC be at an angle at the circumference, BOC an angle at the centre standing on the same arc BEC:

then shall the angle BAC be half the angle BOC.

First let the centre O lie on AB, one of the arms of the angle BAC.

III. Def. 1. 1. 7.

therefore the angle OCA is equal to the angle OAC; therefore the angle OAC is equal to half the sum of the angles OAC, OCA;

I. 25.

but the angle BOC, being the exterior angle of the triangle OAC, is equal to the sum of the angles OAC, OCA; therefore the angle BAC is half the angle BOC, standing on the arc BEC.

Next, let O lie within the angle BAC.

E

Join AO, and produce AO to meet the circumference at D.
Then, as before, the angle BAO is half the angle BOD,

and the angle OAC is half the angle DOC;

therefore the whole angle BAC is half the whole angle BOC standing on the arc BEC.

Again, let O lie without the angle BAC :

B

As before, join AO, and produce AO to meet the circumference at D,

then, the angle BAO being half the angle BOD, and the angle OAC half the angle DOC,

therefore the remaining angle BAC is half the remaining angle BOC. Q.E.D.

* Ex. 41. Two chords AB, CD intersect at a point E within the circle. Shew that the angle AEC is half the sum of the angles at the centre standing on the arcs AC, BD.

* Ex. 42. Two chords AB, CD, produced intersect at a point E without the circle. Shew that the angle AEC is half the difference of the angles at the centre standing on the arcs AC, BD.

THEOR. 16. Angles in the same segment are equal to one another.

Let BAC, BDC be angles in the same segment BADC;

then shall the angle BAC be equal to the angle BDC.

Take O the centre of the circle; join OB, OC.

Then the angles BAC, BDC are angles standing on the same arc, namely the arc BEC which is conjugate to the arc BADC, and therefore each is half of the angle BOC on the arc BEC: III. 15. therefore the angle BAC is equal to the angle BDC.

Ax. h.

Q.E.D.

COR. I. The angle subtended by the chord of a segment at a point within it is greater than, and that at a point outside the segment and on the same side of the chord as the segment is less than, the angle in the segment.

Let D be a point within the segment BAC :

then shall the angle BDC be greater than the angle in the segment BAC.

Produce BD to meet the circumference at E, and join EC.