From O as a center, with OE as a radius, describe the circle EFG. Then, EFG is the circle required.

Proof. Const., o lies in the bisectors of A and B ;

.. § 134,

o is equidistant from AB, AC, and BC.

Q.E.F.

Hence, a circle described from o as a center, with a radius equal to OE, touches AB, AC, and BC.

That is, § 190, the circle EFG is inscribed in ▲ ABC.

Proposition XXXV

250. Problem. To divide a straight line into equal parts.

Datum: Any straight line, as AB.

Required to divide AB into equal parts.

Solution. From A draw a line of indefinite length, as AD, making any convenient angle with AB.

On AD measure off in succession equal distances corresponding in number with the parts into which AB is to be divided.

From the last point thus found on AD, as C, draw CB, and from each point of division on AC draw lines | CB and meeting AB. These lines divide AB into equal parts.

Q.E.F.

Proof. By the student. SUGGESTION.

Refer to § 157.

Ex. 231. If the sides of a central angle of 35° intercept an arc of 75cm, what will be the length of an arc intercepted by the sides of a central angle of 80° in the same circle?

Ex. 232. AB and CD are diameters of the circle whose center is 0; BD is an arc of 116°. How many degrees are there in each angle of the triangles

AOC and DOB?

Ex. 233. If a circle is circumscribed about a triangle ABC, and perpendiculars are drawn from the vertices to the opposite sides and produced to meet the circumference in the points D, E, and F, the arcs EF, FD, and DE are bisected at the vertices.

Proposition XXXVI

251. Problem. To find the center of a circle.

Datum: Any circle, as ABD.

Required to find the center of ABD.

Solution.

Draw any two non-parallel chords, as AB and CD.

Draw the perpendicular bisectors of AB and CD, and produce them until they intersect, as in 0.

Then, is the center of the circle.

Proof. By the student.

SUGGESTION. Refer to § 201.

Ex. 234. To circumscribe a circle about a given triangle.

Ex. 235. AB is a chord of a circle and AC is a tangent at A; a secant parallel to AB, as EFD, cuts AC in E and the circumference in F and D ; the lines AF, AD, and BD are drawn. Prove that the triangles AEF and ADB are mutually equiangular.

Proposition XXXVII

252. Problem. Through a given point to draw a tangent to a given circle.

Data: A circle whose center is 0, and

any point, as A.

Required to draw through 4 a tangent to the circle.

Solution.

Case I. When A is on the circumference.

Draw the radius OA.

At A draw EF LOA.

Then, EF is the tangent required.

Proof. By the student.

Case II. When A is without the circumference.

SUGGESTION. Draw the radii OC and OD, and refer to § 227 and § 205.

Proposition XXXVIII

253. Problem. To describe upon a given straight line a segment of a circle which shall contain a given angle.

Data: Any straight line, as AB, and any angle, as r.

Required to describe a segment of a circle upon AB which shall contain Zr.

M

Solution.

Construct ABD equal to ≤r.

Draw FELAB at its middle point.

Erect a perpendicular to DB at B, and produce it to intersect

FE at 0.

From O as a center, with a radius equal to OB, describe a circle. Then, AMB is the segment required.

Proof. Inscribe any angle in segment AMB, as s.

Q.E.F.

SUMMARY

254. Truths established in Book II.

1. Two lines are equal,

α.

If they are radii of the same circle, or of equal circles.

b. If they are diameters of the same circle, or of equal circles.

Ax. 14

Ax. 15

C. If they are chords which subtend equal arcs in the same circle, or in equal circles.

§ 196

d. If they represent the distances of equal chords in the same circle, or in equal circles, from the center.

of equal circles.

e. If they are chords equally distant from the center of the same circle, or

§ 202

f. If they are tangents drawn to a circle from a point without.

g. If they are the limits of two variable lines which constantly remain equal and indefinitely approach their respective limits.

$ 222

§ 202

§ 209

2. Two lines are perpendicular to each other,

a.

If one is a tangent to a circle and the other is a radius drawn to the point of contact.

§ 205

b. If one is the common chord of two intersecting circles and the other is their line of centers.

§ 212

3.

Two lines are unequal,

α. If one is a diameter of a circle and the other is any other chord of that circle.

d. If they are chords of the same circle, or of equal circles, unequally distant from the center.

C. If they represent the distances of unequal chords in the same circle, or in equal circles, from the center.

b. If they are chords of the same circle, or of equal circles, subtending unequal arcs.

§ 192

§ 197

§ 203

§ 204

4. A line is bisected,

a.

C. If it is the common chord of two intersecting circles, by their line of centers.

If it is a chord of a circle, by a radius perpendicular to it. b. If it is a chord of a circle, by a line perpendicular to it and passing through the center.

§ 198

§ 200

§ 212

5. A line passes through a point,

a. If it is the perpendicular bisector of a chord and the point is the center of the circle.

$ 199

b. If it is the line of centers of two tangent circles, and the point is their point of contact,

$ 213

6. Two angles are equal,

a. If they are central angles subtended by equal arcs in the same circle, or in equal circles.

§ 194

b. If they are inscribed in the same segment of a circle, or in equal segments of the same circle, or of equal circles.

7. Two angles are unequal,

§ 226

a. If they are central angles subtended by unequal arcs in the same circle, or in equal circles.

$ 195

8. An angle is measured,

a. If it is a central angle, by the intercepted arc.

b. If it is an inscribed angle, by one half the intercepted arc. c. If it is between a tangent and a chord, by one half the intercepted

§ 224

§ 225

arc.

d. If it is a right angle, by one half a semicircumference.

§ 231

g. If it is between two secants intersecting without the circle, by one half the difference of the intercepted arcs.

f. If it is between a tangent and a secant, by one half the difference of the intercepted arcs.

e. If it is between two intersecting chords, by one half the sum of the intercepted arcs.

§ 232

§ 230

§ 233

§ 234

9. Two arcs are equal,

a. If they are arcs of the same circle, or of equal circles and their extremities can be made to coincide.

§ 193

b. If they subtend equal central angles in the same circle, or in equal circles. $194

d. If they are intercepted on a circumference by parallel lines. 10. Two arcs are unequal,

c. If they are subtended by equal chords in the same circle, or in equal circles.

§ 196

§ 206

a. If they subtend unequal central angles in the same circle, or in equal circles. § 195

b. If they are subtended by unequal chords in the same circle, or in equal circles.

§ 197

11. An arc is bisected,

a. By the radius perpendicular to the chord that subtends the arc. b. By a line through the center perpendicular to the chord.

a. If it is perpendicular to a radius at its extremity.

§ 205