D

182. Theorem.

PROPOSITION XV

An angle between two tangents, a tangent and a secant, or two secants, intersecting without the circumference is measured by one-half the difference of the intercepted arcs.

Cons.

Dem.

Draw BC (Fig. 1) and DC (Fig. 2) and (Fig. 3)

Hint: (for all cases)

X=A+Y

A = X - Y

EXERCISES

130. If, in Fig. 1 above, the arc BFC is two-fifths of the circumference, how many degrees are there in the angles A, X, and Y?

131. If, in Fig. 3 above, the angle Y= 25° and the arc BC= 105°, how many degrees are there in ZA, and in the sum of the arcs BD and EC?

132. Prove that the angle between two tangents is the supplement of the angle between two radii drawn to the points of contact.

Ex. 137. Prove AD = AE Ex. 138. FG is tang. to OEDA

EXERCISES

145. If each base angle of an inscribed isosceles triangle is double the vertical angle, tangents to the circle at the three vertices of the triangle will form an isosceles triangle in which each base angle is one-third the vertical angle.

146. AB is the diameter of a circle and CD a chord perpendicular to AB; E is any point in the arc BC, and AE cuts CD at F. Prove that LEFD == LACE.

147. Two circles are tangent internally, and the radius of one equals the diameter of the other. AB is a common diameter, and CD is a chord of the larger circle drawn through the centre of the smaller, perpendicular to AB and cutting the smaller circumference at E and F. GH is a second chord of the larger circle, perpendicular to CD at E or F. greater segments of the chords CD and GH are equal, segments.

Prove that the also the lesser

148. The bisector of the angle between two tangents passes through the centre of the circle.

149. The bisectors of the angles of a circumscribed quadrilateral are concurrent.

150. The bisectors of the angles of an inscribed triangle meet at D. If AD produced meets the circumference at E, prove that ED = EC.

Prove

151. In a circle whose centre is A are drawn two circles with centres B and C which touch each other and also the outer circumference. that the perimeter ABC equals the diameter of the outer circle.

152. ABC is a triangle inscribed in a circle whose centre is O, and OD is drawn perpendicular to AC. Prove that ZAOD = B.

153. If a circle is inscribed in a right triangle, the sum of the legs equals the hypotenuse plus two radii.

154. AO and BO are radii of a circle perpendicular to each other. AC is a chord intersecting OB at D. If a tangent at C meets OB produced at E, prove that DEC is an isosceles triangle.

155. A circle inscribed in the triangle ABC touches the sides AB, BC, and CA at D, E, and F, respectively. Prove that DEF = 90° A. intersecting circles, and AC and AD Prove that CD passes through B.

156. AB is a common chord of two are the diameters of the two circles.

157. AB and AC are tangents to a circle, and D any point in the greater arc BC. Prove that the sum of the angles ABD and ACD is constant.