DEFINITION.

An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment to the extremities of the straight line which is the base of the segment.

PROPOSITION XIV.

The angles in the same segment of a circle are equal to one another.

Let APB, AQB be angles

in the same segment APQB. The APB shall be = 4

AQB.

Ist. Let the segment be

greater than a semicircle.

Join A and B with the

centre of the O.

Then / ACB is double each of the 2 s APB, AQB ; (III. 13)

L

.. LS APB, AQB are equal to one another.

PROPOSITION XV.

The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

F

Let ABFG be a quadrilateral figure inscribed in the circle ABF.

Then shall the S ABF, AGF be together equal to two rights.

So also shall the 4S BAG, BFG.

Join AF, BG.

.'.

Then the AGF is

but

=

the S.AGB, FGB:;

AGB = L AFB,

they are in the same segment AGFB: (111. 14) and FGB = ▲ FAB,

..they are in the same segment FGAB; (III. 14) .. LAGF the S AFB, FAB.

=

LAGF together with the ▲ ABF is = the 4s AFB, FAB, ABF the threes of ▲ AFB,

=

(I. 24)

the S BAG, BFG are together two right angles.

PROPOSITION XVI.

In a circle, the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

Let APB be a of which C

is the centre,

ACB a diameter,

chord dividing the

and AP a

APB into

B

the segments ABP, AQP.

Then shall the in the semi

circle APB be a right ;

the in the segment ABP

greater than a semicircle < a right ≤ ;

and the in the segment AQP less than a semicircle > a right .

Join A and P with any point Q in the arc AQP;

.. the whole

L BPC = =

LAPC= = L PAC;
L PBC;

(I. 2)

(1.2)

APB = the LS PAC, PBC together;

but the threes of ▲ APB are together = two right ≤ s; (1.24) .. ▲ APB in the semicircle is a right .

Again, .the 4 s APB, ABP are together <two rt. ≤ s, (1.13) APB is a right ;

and

▲ ABP in the

circle is a right 4.

segment ABP greater than a semi

Also, . ABPQ is a quadrilateral figure in a circle,

=

.. LS ABP, AQP are together two rights, (III.15)

.. the

but ABP is < a right <;

AQP in the segment AQP less than a semicircle is a right 4,

PROPOSITION XVII.

If a straight line touches a circle, and from the point of contact a straight line be drawn cutting the circle; the angles which this straight line makes with the tangent shall be equal to the angles in the alternate segments of the circle.

Through A draw the diameter AD; join DF.

Then AFD is a right angle;

=

.. LS FDA, FAD are together a right;

= ¿

T

(III. 16)

and.. DAT; (III. 2)

Again, since S FRA, AQF are together two right

LEMMA.

Circles, whose radii are equal, are equal in all respects; for, if one circle be applied to the other so that their centres coincide, then will their circumferences coincide also.

PROPOSITION XVIII.

In equal circles equal angles stand upon equal arcs.

Let AFM, BHN be equal circles; and let the s FPG, HQK at their centres be equal to one another. The arc FMG shall be equal to the arc HNK.

For let the AFM be applied to the

BHN,

so that P may fall on Q and PF on QH;

then will PG fall on QK,

LFPG: =

L HQK;

and the Oce AFM will fall on the Oce BHN; (Lemma)

.. the arc FMG must coincide with arc HNK,

COR. I. The chord FG will coincide with the chord HK. Also the tangents at F, G will coincide with the tangents at H, K.

COR. 2. In equal circles equal chords cut off equal arcs. For FG being = HK, and also PF = QH and PG = QK, it follows that FPG .. arc FMG = arc HNK.

=

L HQK;

(1.5)