171. Theorem. If two circles are tangent to each other, the point of contact lies on the line of centers. Hypothesis. Circles with centers A and B, respectively, are tangent at C. Conclusion. Clies on AB. Let MN be the common tangent at C. Draw AC and BC. It can be proved that C is on AB if it is proved that ACB is a straight line, which in turn requires the proof that ACB is a straight angle. EXERCISES 1. If two circles are tangent to each other, the distance between their centers equals the sum or difference of their radii. 2. Three circles are drawn each tangent to the other two. The lines of centers are 12 in., 16 in., and 18 in., respectively. Find the radii. 5. If two circles are tangent externally at A and have a common tangent touching them at B and C, respectively, a circle with diameter BC will pass through A. 6. In the figure of Ex. 4, BAC is a right angle. 7. If two equal circles are tangent externally at A, and a line is drawn through A intersecting the circles again at B and C, respectively, the chords AB and AC are equal. SUGGESTION.-Draw the line of centers. 8. A common internal tangent of two equal circles bisects the line of centers. 9. If two equal circles intersect, the common chord bisects the line of centers. 10. If two circles meet on their line of centers, they are tangent. 11. If two circles are externally tangent, and a line is drawn through the point of contact, terminating in the circles, the diameters drawn through the extremities are parallel. SUGGESTION. - Draw the line of centers. Show that the proof holds if the circles are internally tangent. 12. If two circles intersect and a straight line is drawn parallel to their common chord and cutting both circles, the segments of the line intercepted between the circles are equal; that is, in the figure, EC FD. = E M N B 14. If a triangle ABC is formed by the intersection of three tangents to a circle, two of which, AD and AE, are fixed, while the third, BC, touches the circle at a variable point F on the arc DE, prove that the perimeter of the triangle ABC is constant and equal to AD+AE. 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 radius r and tangent externally or internally to a given circle? Prove the answer. 19. The adjoining figure is used much in different decorative designs, such as ornamental church windows. Arcs AB, AC, and BC are drawn with centers at C, B, and A, the vertices of an equilateral triangle. Arcs ADFC, BDEC, and AEFB are semicircles. Where are the centers A of these semicircles? C 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 N are tangent at A. The lines PM and PN, joining any point P in the common tangent at A to the centers, cut the circles at B and C, respectively. Prove that PB PC. BC A 21. This figure shows a Gothic window. It contains two arcs, and AC, drawn with centers at A and B, respectively, vertices of equilateral ▲ ABC. It contains four smaller arcs, AM, DM, DN, and BN. The circle with center O is drawn with radius equal to AE. A EDF B 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 A B B о C The point of contact A is to each other at A form a compound curve. the transition point from one circle to the other. Compound curves are used in the construction of railroads where the winding track must be made to conform to the physical features of the land. They also secure "easement," preventing a train from lurching when it comes to a curve. The curve ABC of a railroad track is composed of two arcs, AB and BC. The B A M 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 BM. 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. A B D E 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 Draw de such moldings. signs of other moldings in which compound curves are used. B 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. Prove that AD = DC = DB =√R × r. SUGGESTION. - Draw MD and ND. Prove ▲ MCD ~ ▲ NCD. 172. Theorem. Two parallel lines intercept equal arcs on a circle. 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. 2. Draw diameter MN. Hyp. |