222. THEOREM. The angle formed by two secants, two tangents, or a tangent and a secant, meeting outside a circle, is measured by one half the difference of the intercepted arcs.

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Outline of Proof: In each case the given angle is equal to 1, and the arc which measures 1 is the difference between two arcs, one of which is the larger of the two intercepted arcs and the other is equal to the smaller. For instance, in the first figure,

1. If (in left figure, § 222) ▲ A 17° and EB = 25°, find DF. 2. If ZA = 37° (in middle figure), find the arcs into which the points B and E divide the circle.

3. With a given radius construct a circle passing through a given point. How many such circles can be drawn? What is the locus of the centers of all such circles?

4. Draw a circle passing through two given fixed points. How many such circles are there? What is the locus of the centers of all such circles?

5. Construct a circle having a given radius and passing through two given points. How many such circles can be drawn? Is this construction ever impossible? Under what conditions is only one such circle possible?

224. PROBLEM. To construct a circle through three fixed points not all in the same straight line.

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Given three points A, B, C not in the same straight line.
To construct a circle passing through them.

Construction. Let the student give the construction and proof in full. (See § 132.)

225. Definition. The circle OA in § 224 is said to be circumscribed about the triangle ABC and the triangle is said to be inscribed in the circle.

1. In the construction of § 224 why do DM and EN meet?

2. Why cannot a circle be drawn through three points all lying in the same straight line? Make a figure to illustrate this.

3. Show that an angle inscribed in an arc is greater than or less than a right angle according as the arc in which it is inscribed is less than or greater than a semicircle.

4. Prove that the bisectors of the angles of an equilateral triangle pass through the center of the circumscribed circle.

5. Draw a circle tangent to two fixed lines. How many such circles are there? What is the locus of their centers? Is the point of intersection part of this locus? Discuss fully.

6. Show that not more than one circle can be drawn through three given points, and hence that two circles which coincide in three points coincide throughout.

227. PROBLEM.

To construct a circle tangent to each

of three lines, no two of which are parallel and not all of which pass through the same point.

To construct a circle tangent to each of these lines. Construction. Since no two of the lines are I, let 7 and l1⁄2 meet in A, l1⁄2 and l。 in B, and l。 and l1 in D, where A, B, and D are distinct points.

Draw the bisectors of A and B and let them meet in point C.

Then C is the center of the required circle. (See § 131.) Give the proof in full.

228. Definitions. The circle in the construction of § 227 is said to be inscribed in the triangle ABD.

Three or more lines which all pass through the same point are called concurrent.

current.

229.

Hence the lines l1, 2, lg are not con- .

EXERCISES.

1. Why is the construction of § 227 impossible if 11, 12, and 13 are

concurrent?

2. If two of the lines are parallel to each other, show that the construction is possible. How many tangent circles can be constructed in this case? Draw a figure and give the construction and proof in full.

3. Is the construction possible when all three lines are parallel? Why?

4. If two sides of the triangle are produced, as AB and AD in the figure of § 227, construct a circle tangent to the side BD and to the prolongations of the sides AB and AD.

This is called an escribed circle of the triangle.

5. How many circles can be constructed tangent to each of three straight lines if they are not concurrent and no two of them are parallel?

6. Draw a triangle and construct its inscribed and circumscribed circles and its three escribed circles.

230. PROBLEM. From a given point outside a circle to draw a tangent to the circle.

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Given O CA and an outside point P.

To construct a tangent from P to the circle.

Construction.

Draw CP. On CP as a diameter con

struct a circle, cutting the given circle in the points A and B.

1. If in the figure of § 230 the point P is made to move towards the circle along the line PC until it finally reaches the circle, while PA and PB remain tangent to the circle, describe the motion of the points A and B and also of the lines PA and PB. How does this agree with the fact that through a point on the circle there is only one tangent to the circle?

2. Can a tangent be drawn to a circle from a point inside the circle? Why?

3. Show that the line connecting a point outside a circle with the center bisects the angle formed by the tangents from that point.

4. Why are not more than two tangents possible from a given point to a circle?

5. The two tangents which can be drawn to a circle from an exterior point are equal.

6. In a right triangle the hypotenuse plus the diameter of the inscribed circle is equal to the sum of the two legs of the triangle.

7. If an isosceles triangle inscribed in a circle has each of its base angles double the vertex angle, and if tangents to the circle are drawn through the vertices, find the angles of the resulting triangle.

8. If the angles of a triangle ABC inscribed in a circle are 64°, 72°, and 44°, find the angles of the triangle formed by the tangents to the circle at the points A, B, and C.

SUMMARY OF CHAPTER II.

1. Make a list of all the definitions involving the circle.

2. State the theorems on the measurement of angles by intercepted arcs.

3. State the theorems involving equality of chords, central angles, and intercepted arcs.

4. State the theorems on the tangency of straight lines and circles.

5. State the theorems involving the tangency of two circles.

6. Make a list, to supplement that in the summary of Chapter I, of ways in which two angles or two line-segments may be proved equal. 7. State the ways in which two arcs of the same or equal circles may be proved equal.

8. State the problems of construction given in Chapter II.

9. Explain what is meant by saying that a central angle is measured by its intercepted arc.

10. State some of the important applications of Chapter II. (Return to this question after studying those which follow.)