PROBLEMS AND APPLICATIONS.

1. If the distance between the centers of two circles is equal to the sum of their radii, how are the circles related? Construct and prove. 2. If the distance between the centers of two circles is equal to the difference of their radii, how are the circles related? Construct and prove?

3. If the distance from the center of a circle to a straight line is equal to the radius, how is the line related to the circle? Construct and prove.

4. Given two circles having the same center, construct a circle tangent to each of them. Can more than one such circle be constructed? What is the locus of the centers of all such circles?

5. Prove the converse of the theorem in § 209.

6. The straight line joining the centers of two intersecting circles bisects their common chord at right angles.

7. A line tangent to each of two equal circles is either parallel to the segment joining their centers or else it bisects this segment.

9. In the figure A, B, C, D are the vertices of a Show how to construct the entire figure. What semicircles are tangent to each other?

square.

This construction occurs frequently in

designs for tile flooring. See accompanying figure. This is from a Roman mosaic.

10. Given two parallel lines BE and AD, to con- B struct arcs which shall be tangent to each other and one of which shall be tangent to BE at B and the O other tangent to AD at A.

SOLUTION. Draw AB and bisect this segment at C; construct1 bisectors of AC and BC. From A and B draws to AD and BE respectively, thus locating the points O and O'.

Prove that O and O' are the centers of the required

arcs.

SUGGESTION. Show that O, C, and O' lie in a straight line and use the theorem of § 209.

This construction occurs in architectural designs and in many other applications. In the accompanying designs pick out all the arcs that C are tangent to each other and also the points of tangency.

Scroll Work.

E

11. On the sides of the equilateral triangle ABC as diameters, semicircles are drawn, as AEFB. Also with A, B, C as centers and AB as radius arcs are drawn,

as AB, BC.

B

Fourth Presbyterian Church, Chicago.

SUGGESTION. If the middle points of the sides of an equilateral triangle are joined, what kind of triangles are formed?

(b) What arcs in this figure are tangent to each other?

(c) Has the figure one or more axes of symmetry?

This figure and the two following occur frequently in church windows and other decorative designs.

11. Construct the design shown in the figure.

SUGGESTION. Divide the diameter AB into six equal parts and construct the three semicircles.

On DC and DC' as bases construct equilateral triangles with vertices O and O'.

With radius equal to CB and centers O and O' construct circles.

A C

D

C B

(a) Prove that OO is tangent to each of the three semicircles. Likewise O 0'.

(b) Erect a 1 to AB at D and prove © O and O' tangent to it. (c) Prove circles with centers at O and O' tangent to each other. (d) Has this figure one or more axes of symmetry?

12. In the figure AB, CD and OD are bisected, and O'O' || AB through E. DO' = DO" = 3 DB. Circles are

constructed as shown in the figure.

(a) If AB is 4 feet, what is the radius of

each circle?

(b) Prove that O O is tangent to ○ O' and also to OO".

SUGGESTION. Show that 00' is the sum of the radii of the two

circles.

(c) Is O' tangent to the arc ACB and also to the line AB? (d) Has this figure one or more axes of symmetry?

13. ABCD is a square. Arcs are constructed with A, B, C, D as centers and with radii each equal to one half the side of the square. The lines AC, BD, MN, and RS are drawn, and the points E, F, G, H are D connected as shown in the figure.

The arc SN is extended to P, forming

a semicircle. The line LP meets SN in M K, and BK meets MN in O'.

(a) Prove that EFGH is a square.

(b) Prove that ▲ KLO' and KPB are

mutually equiangular and each isosceles.

R

H

N

E

A

S

B

(c) Prove that O'K is tangent to FG and to SN.

(d) How many axes of symmetry has the solid figure?

(e) Show that O'K is tangent to ŔŇ by drawing O'C and fold

ing the figure over on the axis of symmetry MN.

210. THEOREM. An angle inscribed in a circle is measured by one half the intercepted arc.

Given DBA inscribed in CB.

To prove that DBA is measured by AD. Proof: (1) If one side, as BD, is a diameter. Draw the radius CA. Show that 2 = 1 ≤ 1. But 1 is measured by AD (§ 202).

Hence 2 is measured by AD.

(2) If the center C lies within the angle. Draw the diameter BE.

Now DBA = 21+ 22.

Complete the proof.

(3) If the center C lies outside the angle.

Draw BE and use the equation ≤ DBA = ≤1 – Z 2.

211. It follows from § 210 that if in equal circles two inscribed angles intercept equal arcs, they are equal; and conversely, that if equal angles are inscribed in equal circles, they intercept equal arcs.

1. If the sides of two angles BAD and BA'D pass through the points B and D on a circle, and if the vertex A is on the minor arc BD and A' is on the major arc BD, find the sum of the two angles.

2. In Ex. 1 if the points B and D remain fixed while the vertex A of the angle is made to move along the minor arc of the circle, what can be said of the angle A? What if it moves along the major arc?

213. THEOREM.

The locus of the vertices of all right

triangles on a given hypotenuse is a circle whose diame

whose diameter is AB, APB = rt. Z.

(Why?)

(2) If AP'B is any right triangle with AB as hypotenuse, then AC = CB = CP'. (See Ex. 27, p. 82.)

1. The centers of all circles of fixed radius tangent to a fixed line.

2. The centers of all circles tangent to two parallel lines.

3. The centers of all circles tangent to both sides of an angle.

4. The centers of all circles tangent to a given line at a given point. Is the given point a part of this locus?

5. The vertices of all triangles which have a

common base and equal altitudes.

6. The middle points of all chords through a fixed point on a circle. Use Ex.

parallel lines.

8, § 208, and then § 213.

7. The points of intersection

of the diagonals of trapezoids

formed by the sides of an isosceles triangle and lines parallel to its base.

8. Two vertices of a triangle slide along two What is the locus of the third vertex if the triangle is fixed in size and shape?

9. ABCD is a parallelogram all of whose sides are of fixed length. The side AB is fixed in position. Find the locus of the middle points of the remaining three sides.