EXERCISES.

119. If a tangent and a chord of a circle are parallel, prove that the arcs intercepted between the tangent and chord are equal.

120. Upon a given base, construct an isosceles triangle, in which the sum of two equal sides shall equal a given line.

121. All angles inscribed in the same segment are equal.

122. Prove that an angle inscribed in

A

a semicircle is a right angle.

To prove that the angle BAC is a right angle.

C

123. Demonstrate Prop. XXII by another method.

SUG. Draw B E. By what arc is LAB E measured? By what arc is BEC measured? Then, by what arc must A be measured?

124. In the same circle, or in equal circles, an angle inscribed in the smaller of two segments is larger than an angle inscribed in the larger segment.

Prove that the angle A E B is larger than the angle C O D.

B

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

arcs.

126. The segments of a straight line

intercepted by concentric circles are

equal.

Prove A B equals CD.

Concentric circles are circles having the same center.

B

PROPOSITION XXXVII. PROBLEM.

197. Through a given point, to draw a tangent to a given circle.

CASE I. When the point is on the circumference.

Let A represent the given point in the circumference of the given circle.

To draw a tangent to the circle O, through the point A. SUG. 1. What relation does a tangent bear to the radius drawn to the point of contact?

SUG. 2. Give the method of making the construction in this case.

CASE II.

When the given point is without the circle.

4

Let A represent the given point without the circle O.

To draw a tangent to the circle O, through the point A.

SUG. 1. Connect O and A.

SUG. 2. If the required point of contact were joined

both to O and A, what kind of an

would be formed?

SUG. 3. What is the locus of the vertex of the in a ▲ whose base is O A? See Art. 177, and Exs. 121, 122. SUG. 4. Give, now, complete directions for finding the point of contact, and hence for constructing the tangent.

PROPOSITION XXXVIII.

PROBLEM.

198. To circumscribe a circle about a given triangle.

B

Let A B C represent the given triangle.

To circumscribe a circle about the triangle ABC. SUG. 1. The problem is to find the center of a O whose circumference passes through A, B and C; i. e., to find a point equally distant from A, B and C.

SUG. 2. See Art. 158, and then give complete directions for circumscribing a O about the ▲ A B C.

QUERY. How many circles can be circumscribed about a triangle ?

PROPOSITION XXXIX. PROBLEM.

199. To inscribe a circle in a given triangle.

Let A B C represent the triangle.

To inscribe a circle in the triangle A B C.

SUG. 1. If a O can be inscribed in the A, the center of the must be equally distant from the three sides. SUG. 2. Is there such a point? See exercise 71.

SUG. 3. How may this point be found?

SUG. 4. Give complete directions for inscribing a circle in a given A.

QUERY. How many circles can be inscribed in a given triangle? Why?

PROPOSITIONS IN CHAPTER II.

PROPOSITION I.

Two circles are equal if the radius of one equals the radius of the other.

PROPOSITION II.

A diameter divides a circle into two equal parts.

PROPOSITION III.

In the same circle, or in equal circles, equal angles at the center intercept equal arcs at the circumference.

PROPOSITION IV.

CONVERSE OF PROP. III. In the same circle, or in equal circles, equal arcs subtend equal angles at the center.

PROPOSITION V.

In the same circle, or in equal circles, chords which subtend equal arcs are equal.

PROPOSITION VI.

Converse OF PROP. V. In the same circle, or in equal circles, arcs which are subtended by equal chords are equal.

PROPOSITION VII.

In the same circle, or in equal circles, two chords which subtend unequal arcs are unequal, that chord being greater which subtends the greater arc.

PROPOSITION VIII.

CONVERSE OF PROP. VII. In the same circle, or in equal circles, two arcs which are subtended by unequal chords are unequal, that arc being greater which is subtended by the greater chord.

PROPOSITION IX.

A radius which is perpendicular to a chord bisects the chord and its subtended arc.