15. Three equal circles are tangent to each other. Prove that the three common internal tangents meet at a point which is equidistant from the three points of contact.

SUGGESTION. - Let two tangents meet, and join their point of intersection to the third point of contact. Also join this point of intersection to the centers, and draw the radii to the points of contact.

16. Show that the locus of the centers of all circles tangent to a given circle at a given point is the straight line determined by the given point and the center of the given circle.

17. Circles are drawn tangent to a given line at the same point of it. From another point of the line tangents are drawn to all of the circles. Prove that the locus of the points of contact of these tangents is a circle.

18. What is the locus of the centers of

all circles with a given radiusr and tangent externally or internally to a given circle? Prove the answer.

19. The adjoining figure is used much in different decorative de

Prove that each semicircle is tangent to two of the arcs, AB, AC, and BC.

Make a large drawing of a window similar to this one.

20. Two equal circles with centers M and Nare tangent at A. The lines PM and PN, joining any point Pin the common tangent at A to the centers, cut the circles at Band C, respectively. Prove that

PB = PC.

C

21. This figure shows a Gothic window. It contains two arcs, BC and AC, drawn with centers at A and B, respectively, vertices of equilateral A ABC. It contains four smaller arcs, AM, DM, DN, and BN. The circle with center O is drawn with radius equal

to AE.

AA

M

W

AEDFB

What are the centers and radii of AM, DM, DN, and BN?

How is the center O located?

Prove the circle with center O tangent to AC, BC, DM, and DN.

22. Arcs AB and AC of two circles, respectively, which are tangent

to each other at A form a compound curve. The point of contact A is the transition point from one circle to the other.

Compound curves are used in the construction of railroads where the

center N of BC is on the radius MB of AB. Prove that ABC is a compound curve, or that AB and BC are arcs of tangent circles. SUGGESTION. - Draw a line through B perpendicular to BМ. 23. AB is a straight railroad track. The track of a switch ACDE

is laid out as follows: Arc AC is constructed

with center M. Then MC is produced to N, making CN = MC, and CD drawn with

center N.

Prove that ACD is a compound curve, or that AC and CD are arcs of tangent circles.

N

B

A

24. Compound curves are used in many kinds of molding, as well as in other architectural de

signs.

Explain the construction of the curves in the adjoining drawings of moldings. Make a large drawing of such moldings. Draw designs of other moldings in

which compound curves are used.

25. A railroad Y consists of three tracks, AC, CB, and AB, upon which a train is reversed in direction, by moving as shown by the arrows, backing from C to B. In constructing a Y, AB is a straight track, and point A given. It is required to locate the curves AC and CB with given radii R and r, respectively.

A

D

B

N

C

M

Prove that AD = DC = DB = √R × r.

SUGGESTION.

Draw MD and ND. Prove & MCD ~ △ NCD.

CASE I. When the parallels are a tangent and a secant. Hypothesis. AB is tangent to a circle at M; CD is a secant

parallel to AB.

Conclusion. CM = DM.

Proof. 1. AB is tangent at M; secant CD || AB. Нур. 2. Draw diameter MN.

3. Then MNLAB.

4. ... MN1 CD.

5... CM = DM.

§ 164

§ 34

§ 156

CASE II. When both of the parallels are secants.

Hypothesis.

Secants AB and CD are parallel.

Conclusion. AC = BD.

Suggestion. Draw EF parallel to AB and tangent at a point M. Apply Case I.

CASE III. When both of the parallels are tangents.

Hypothesis. AB is tangent at M and CD tangent at N;

AB || CD.

Conclusion.

Suggestion.

Case I.

Arc MEN = arc MFN.

Draw secant EF parallel to AB. Apply

Write the proof of the theorem in full.

EXERCISES

1. Prove that the two non-intersecting chords which join the extremities of two parallel chords are equal.

2. Prove that the two intersecting chords which join the extremities of two parallel chords are equal.

3. Prove that an inscribed trapezoid is isosceles.

4. Prove that the diagonals of an inscribed trapezoid are equal.

5. Prove, by using Case III of the theorem in § 172, that a diameter of a circle bisects the circle.

6. Prove that if a tangent and a secant intercept equal arcs of a circle, they are parallel.

7. Prove that if two secants which do not intersect within the circle intercept equal arcs, they are parallel.

8. Prove that the opposite sides of an inscribed equilateral hexagon are parallel.

173. The relations of angles and arcs. - If a central angle AOB is divided into unit angles, such as degrees, by draw

ing radii, then AB is divided into an equal

number of equal arcs. Why? If one of these equal arcs is taken as the unit of measure of arcs, then the numerical measure of AB is the same as that of ∠AOB.

This relation may be proved to be equally true when the units of measure are not con

B

A

tained exactly in ∠AOB and AB, that is, when the numerical measures of ∠AOB and AB can be expressed only approximately. The proof is omitted from this course.

In general, if the unit of measure of arcs is chosen as the arc intercepted by a unit central angle,

A central angle has the same numerical measure as its intercepted arc.

174. Corollary. - In the same circle or equal circles, any two central angles have the same ratio as their intercepted arcs.

This follows from the principle in § 173 by definition of ratio. See § 116.

175. Degrees, minutes, and seconds of arc. - An arc which a central angle of one degree intercepts is called a degree of arc. Similarly, the arcs which central angles of one minute and of one second intercept are called a minute of arc and a second of arc, respectively.

Hence, the number of degrees, minutes, and seconds of arc in an arc equals the number of degrees, minutes, and seconds in the central angle which intercepts it.

Thus, a central angle of 62° 15′ 30′′ intercepts an arc of 62° 15′ 30′′.