a.

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.

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

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

§ 202

§ 202

§ 209

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

2. Two lines are perpendicular to each other,

§ 222

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.

3. Two lines are unequal,

§ 212

α.

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

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

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

§ 203

§ 204

4. A line is bisected,

α.

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.

§ 224

$ 225

C.

If it is between a tangent and a chord, by one half the intercepted

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

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.

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

$ 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

SUPPLEMENTARY EXERCISES

Ex. 236. ABC is an inscribed isosceles triangle; the vertical angle C is three times each base angle. How many degrees are there in each of the arcs AB, AC, and BC?

Ex. 237. If a hexagon is circumscribed about a circle, the sums of its alternate sides are equal.

other are inter

Prove that the

Ex. 238. Two radii of a circle at right angles to each sected, when produced, by a line tangent to the circle. tangents drawn to the circle from the points of intersection are parallel to each other.

Ex. 239. Two circles are tangent to each other externally and each is tangent to a third circle internally. Show that the perimeter of the triangle formed by joining the three centers is equal to the diameter of the exterior circle.

Ex. 240. From two points, A and B, in a diameter of a circle, each the same distance from the center, two parallel lines AE and BF are drawn toward the same semicircumference, meeting it in E and F. Show that EF is perpendicular to AE and BF.

Ex. 241. Two circles are tangent externally at A. BC is a tangent to the two circles at B and C. Prove that the circumference of the circle described on BC as a diameter passes through A.

Ex. 242. OA is a radius of a circle whose center is 0; B is a point on a radius perpendicular to OA; through B the chord AC is drawn; at Ca tangent is drawn meeting OB produced in D. Prove that CBD is an isosceles triangle.

SUGGESTION. Draw a tangent at A.

Ex. 243. Through a given point P without a circle whose center is O two lines PAB and PCD are drawn, making equal angles with OP and intersecting the circumference in A and B, C and D, respectively. Prove that AB equals CD, and that AP equals CP.

SUGGESTION. Draw OM and ON perpendicular to AB and CD, respect

ively.

Ex. 244. If the angles at the base of a circumscribed trapezoid are equal, each non-parallel side is equal to half the sum of the parallel sides.

Ex. 245. If a circle is inscribed in a right triangle, the sum of the diameter and the hypotenuse is equal to the sum of the other two sides of the triangle.

Ex. 246. Any parallelogram which can be circumscribed about a circle is equilateral.

Ex. 247. AB and CD are perpendicular diameters of a circle; E is any point on the arc ACB. Then, D is equidistant from AE and BE.

Ex. 248. If two equal chords of a circle intersect, their corresponding segments are equal.

Ex. 249. If the arc cut off by the base of an inscribed triangle is bisected and from the point of bisection a radius is drawn and also a line to the opposite vertex, the angle between these lines is equal to half the difference of the angles at the base of the triangle.

Ex. 250. The two lines which join the opposite extremities of two parallel chords intersect at a point in that diameter which is perpendicular to the chords.

Ex. 251. If a tangent is drawn to a circle at the extremity of a chord, the middle point of the subtended arc is equidistant from the chord and the tangent.

Ex. 252. A line is drawn touching two tangent circles. Prove that the chords, that join the points of contact with the points in which the line through the centers meets the circumferences, are parallel in pairs.

Ex. 253. Two circles intersect at the points A and B; through A a secant is drawn intersecting one circumference in C and the other in D; through B a secant is drawn intersecting the circumference CAB in E and the other circumference in F. Prove that the chords CE and DF are parallel.

SUGGESTION. Refer to Ex. 198 and 201.

Ex. 254. The length of the straight line joining the middle points of the non-parallel sides of a circumscribed trapezoid is equal to one fourth the perimeter of the trapezoid.

Ex. 255. A quadrilateral is inscribed in a circle, and two opposite angles are bisected by lines meeting the circumference in A and B. Prove that AB is a diameter.

Ex. 256. The centers of the four circles circumscribed about the four triangles formed by the sides and diagonals of a quadrilateral lie on the vertices of a parallelogram.

Ex. 257. If an equilateral triangle is inscribed in a circle, the distance of each side from the center is equal to half the radius of the circle.

Ex. 258., The vertical angle of an oblique triangle inscribed in a circle is greater or less than a right angle by the angle contained by the base and the diameter drawn from the extremity of the base.

Ex. 259. If from the extremities of any diameter of a given circle perpendiculars to any secant that is not parallel to this diameter are drawn, the less perpendicular is equal to that segment of the greater which is contained between the circumference and the secant.

Ex. 260. Two circles are tangent internally at A, and a chord BC of the larger circle is tangent to the smaller at D. Prove that AD bisects the angle CAB.

SUGGESTION.

Draw AT, the common tangent of the circles.

Ex. 261. The tangents at the four vertices of an inscribed rectangle form a rhombus.

Ex. 262. If a line is drawn through the point of contact of two circles which are tangent internally, intersecting the circle whose center is A at C, and the circle whose center is B at D, AC and BD are parallel.

Ex. 263. If lines are drawn from the center of a circle to the vertices of any circumscribed quadrilateral, each angle at the center is the supplement of the central angle that is not adjacent to it.

Ex. 264. Three circles are tangent to each other externally at the points A, B, and C. From A lines are drawn through B and C meeting the circumference which passes through B and C at the points D and E. Prove that DE is a diameter.

Ex. 265. If an angle between a diagonal and one side of a quadrilateral is equal to the angle between the other diagonal and the opposite side, the same will be true of the three other pairs of angles corresponding to the same description, and the four vertices of the quadrilateral lie on a circumference.

Ex. 266. Let the diameter AB of a circle be produced to C, making BC equal to the radius; through B draw a tangent, and from C draw a second tangent to the circle at D, intersecting the first at E; AD and BE produced meet at F. Prove that DEF is an equilateral triangle.

SUGGESTION. Draw a line from the center to E.

Ex. 267. If from any point without a circle tangents are drawn, the angle contained by the tangents is double the angle contained by the line joining the points of contact and the diameter drawn through one of them.

Ex. 268. The lines, which bisect the vertical angles of all triangles on the same base and on the same side of it, and having equal vertical angles, meet at the same point.

Ex. 269. AB and AC are tangents at B and C respectively, to a circle whose center is 0; from D, any point on the circumference, a tangent is drawn, meeting AB in E and AC in F. Prove that angle EOF is equal to one half angle BOC.

Ex. 270. A circle whose center is O is tangent to the sides of an angle ABC at A and C; through any point in the arc AC, as D, a tangent is drawn, meeting AB in E, and CB in F. Prove (1) that the perimeter of the triangle BEF is constant for all positions of D in AC; (2) that the angle EOF is constant.

Ex. 271. If AE and BD are drawn perpendicular to the sides BC and AC, respectively, of the triangle ABC, and DE is drawn, the angles AED and ABD are equal.

SUGGESTION. Describe a circle passing through A, D, and B.