Proposition 15. Theorem.

243. An angle formed by a tangent and a chord from the point of contact is measured by one-half the intercepted arc. Hyp. Let AC be a tangent to the

OBHE at B, and BD any chord of the

E

H

BE is a diameter of the O.

A

B

The to a tangent at the pt. of contact passes through the centre

ABD is measured by

are BED.

Q.E. D.

Similarly, 244. SCH. This proposition is a particular case of Prop. 14. Thus, let the side BD remain fixed, while the side BH turns about B, as in (28), until it becomes the tangent BC at the point B. In every position of the chord BH, the inscribed angle HBD is measured by half the intercepted arc HD. Therefore, when the chord BH becomes the tangent BC, the angle CBH is measured by half the arc BHD.

EXERCISES.

1. If the angle BAC at the circumference of a circle be half that of an equilateral triangle, prove that BC is equal to the radius of the circle.

2. If a hexagon be inscribed in a circle, show that the sum of any three alternate angles is four right angles.

3. If two circles intersect in the points A, B, and any two lines ACD, BFE, be drawn through A and B, cutting one of the circles in the points C, E, and the other in the points D, F, the line CE is parallel to DF.

Proposition 16. Theorem.

245. An angle formed by two chords which intersect within a circle, is measured by one-half the sum of the arcs intercepted between its sides and between its sides produced.

A D

E

Hyp. Let BD, CE be two chords intersecting at A within the BCDE.

BAC is measured by (arc BC + arc DE).

To prove
Proof. Join BE.

The ext. of a ▲ equals the sum of the opp. int. ≤s (98).

ZAEB is measured by arc BC,

But

1. If arc BC = 84° and CAD is a rt. ≤, how many degrees are there in the arc DE ?

2. The sides of a quadrilateral touch a circle, and the straight lines drawn from the centre of the circle to the vertices cut the circumference in A, B, C, D: show that AC, BD, which intersect inside the circle, are at right angles to each other.

Proposition 17. Theorem.

246. An angle formed by two secants which intersect without a circle, is measured by one-half the difference of the intercepted arcs.

Hyp. Let AC, AB be two secants intersecting at A without the BCED. ZA is measured by

To prove

(arc BC

- arc DE).

Proof. Join BE.

BECA + ZB. (98)

...ZA BEC-B.

BEC is measured by arc BC,

But

and

B is measured by arc DE. (238)

D

B

.. A is measured by (arc BC arc DE).

Q.E.D.

247. SCH. Prop. 14 may be considered as a special case of Props. 16 and 17 by conceiving AB in (245) and (246) to move parallel to its present position until D reaches E. When D reaches E, the arc DE becomes zero, and BAC becomes an inscribed angle, measured by half its intercepted

arc.

EXERCISES.

1. If arc BC = 80° and B = 14°, find the number of degrees in the angle A.

2. A, B, C are three points on the circumference of a circle, the bisectors of the angles A, B, C meet in D, and AD produced meets the circle in E: prove that ED EC.

3. If a quadrilateral be described about a circle, the angles at the centre subtended by the opposite sides are supplemental.

Proposition 18. Theorem.

248. An angle formed by a tangent and a secant is measured by one-half the difference of the intercepted arcs.

E

H

Hyp. Let AC, AB be a tangent and a secant intersecting at A.

-

To prove A is measured by (arc BHE arc DE).
Proof. Join BE.

249. COR. The angle formed by two tangents is measured by one-half the difference of the intercepted arcs.

EXERCISES.

1. Two tangents AB, AC are drawn to a circle; D is any point on the circumference outside the triangle ABC: show that the sum of the angles ABD and ACD is constant.

2. If a variable tangent meets two parallel tangents it subtends a right angle at the centre.

QUADRILATERALS.

Proposition 19. Theorem.

250. If the opposite angles of a quadrilateral are supplementary, the quadrilateral may be inscribed in a circle. Hyp. Let ABCD be a quadrilateral in

which

ZB+D=2 rt. s.

To prove the pts. A, B, C, D are in the same O.

Proof. Through the three pts. A, B, C describe a O.

B

A

E

will cut AD, or AD produced, at some other pt. than D. Let E be this pt. Join EC.

Because the quadrilateral ABCE is inscribed in a O,

The opp. Ls of an inscribed quad. are supplementary (242).

.. the circle which passes through A, B, C, must pass through D.

Q. E. D.

251. DEF. Points which lie on the circumference of a circle are called concyclic.

A cyclic quadrilateral is one which is inscribed in a

circle.

EXERCISE.

If two opposite sides of a cyclic quadrilateral be produced to meet, and a perpendicular be let fall on the bisector of the angle between them from the point of intersection of the diagonals: prove that this perpendicular will bisect the angle between the diagonals.