equal and angle A is greater than angle C. Which is the longer side, AD or CD? Give the reason for your answer.

637. Prove that in a convex quadrilateral the angle between the bisectors of two adjacent angles is one half the sum of the other two angles.

638. An arc of a certain circle is 100 ft. long, and subtends an angle of 25° at the center. Compute the radius of the circle correct to three significant figures.

639. Three successive vertices of a regular octagon are A, B, C, respectively. If the length AB is a, compute the length AC.

640. The areas of similar segments of circles are proportional to the squares of their radii.

641. Given a circle whose radius is 16. Find the perimeter and the area of the regular inscribed octagon.

642. Two circles intersect at points A and B. Through A a variable secant is drawn, cutting the circles in C and D. Prove that the angle CBD is constant for all positions of the secant.

643. Let A and B be two fixed points on the circumference of a given circle and P and Q the extremities of a variable diameter of the same circle. Find the locus of the point of intersection of the straight lines AP and BQ.

644. Prove that in any right triangle the line drawn from the right angle to the middle of the hypotenuse is equal to one half the hypotenuse.

645. The area of a regular decagon is 108 sq. in. Find the radius of the circumscribed circle.

646. Two secants are drawn from the same point to the same circle. The external segment of the first is 5 in. and its internal segment is 19 in. The internal segment of the other secant is 7 in. Find the length of the second secant.

647. On the diameter AB of a circle mark a point P. Through P draw the chord CPD at right angles to AB. Prove that if AP, BP, CP, and DP be taken as diameters of circles, the sum of the areas of the four circles is equal to the area of the original circle.

648. To construct a rectangle, having given the perimeter and the diagonal.

649. Find how far from the base of a triangle of altitude a lines parallel to the base must be drawn to divide the area of the triangle into three equal parts.

650. Prove that two triangles are similar if the sides of one are respectively parallel to the sides of the other.

651. Derive the numerical value of π.

652. Of all triangles having the same base and equal areas, the isosceles triangle has the minimum perimeter.

653. How high is a tree which casts a shadow 70 ft. long, when a man 6 ft. high casts a shadow 8 ft. long?

654. Construct a triangle, given the base, vertical angle, and median drawn to the base.

655. State six propositions concerning parallel lines and prove any one of them.

656. An interior angle of an equiangular polygon is 150°. Find the number of sides.

657. Three cylindrical barrels, diameter of each being 20 in., are placed in a pile with axes horizontal so that each just touches the other two. Find the height of the pile, and the length of the shortest rope to go over the pile and touch the floor on each side.

658. AB is the diameter of a circle of radius 2 in., and AC is a chord such that BAC is 30°. Find area and perimeter of BACB correct to two decimal places.

659. If from a fixed point D, within a triangle ABC, lines are drawn to all points in the perimeter of the triangle, what is the locus of the middle points of those lines?

660. Show that if the radius of a circle is a, the side of the regular inscribed decagon is (5-1) and the side of the regu

661. The base of a triangle is 32 ft. and its height is 20 ft. What is the area of the triangle formed by drawing a line parallel to the base 5 ft. from the vertex?

662. The sides of a triangle are 5, 12, 13. Find the segments into which each side is divided by the bisector of the opposite angle.

663. The sides of a triangle are a, b, c, and the area is k. What is the radius of the inscribed circle?

664. A and B are fixed points, AC is drawn in any direction, and BP is drawn perpendicular to AC, meeting it at P. What is the locus of P?

665. Upon a line about 11⁄2 in. long construct a segment to contain an angle of 60°.

666. Three circles of radius a touch each other, and another circle is circumscribed about them. Find its radius, circumference, and

area.

=

667. AB is a fixed line, angle ACB 45°. Construct the locus of C.

668. Two sides of a triangle are a and b, the included angle 135°. What is the square of the third side c?

669. The lengths of the circumferences of two concentric circles differ by 6 in. Compute the width of the ring to three significant figures.

670. Prove the area of the triangle formed by joining the middle point of one of the nonparallel sides of a trapezoid to the extremities of the opposite side equals one half the trapezoid.

671. The three sides of a triangle are 4 ft., 13 ft., and 15 ft. long. Show that the altitude upon the side of length 15 is 3.2.

secant.

672. Through the vertex A of the parallelogram ABCD draw a Let this line cut diagonal BD in E, and the sides BC, CD (or these sides produced) in F and G, respectively. Prove that AE is a mean proportional between EF and EG.

673. Let ABC be an equilateral triangle, and on the sides AB, BC, CA, lay off AD, BE, CF, each equal to one third AB, and join the points D, E, F, with one another. Prove that the triangle DEF is equilateral, and that its sides are respectively perpendicular to the sides of the given triangle.

674. On the circumference of a circle take two points subtending a right angle at the center, and a third point on the arc between these two. Prove that the perimeter of the triangle formed by the tangents at these three points is equal to the diameter of the circle.

675. An indefinite straight line moves in such a way that it always passes through at least one vertex of a given square, but

never crosses the square. What is the locus of the foot of the perpendicular dropped on the moving line from the center of the square? Describe the locus accurately and prove the correctness of your

answer.

676. Prove the correctness of the following construction for bisecting an angle ABC; upon AB produced beyond B take BD equal to BC and draw a line through B parallel to DC.

677. Show how to construct a chord through a given point A within a circle, so that the extremities of the chord shall be equidistant from another point B.

678. A rod of length a is free to move within a semicircular area of radius a. Describe accurately the boundary of the region within which the middle point of the rod will always be found.

679. A roadway 60 ft. wide is cut through the middle of a circular field 120 ft. in diameter. Compute the area of the remainder of the field correct to 1 per cent of its value.

680. The radii of two circles are 1 in. and √3 in., respectively, and the distance between their centers is 2 in. Compute their

common area to three significant figures.

681. Determine a point P without a given circle so that the sum of the tangents from P to the circle shall be equal to the distance from P to the farthest point of the circle.

682. The image of a point in a mirror is, apparently, as far behind the mirror as the point itself is in front. If a mirror revolves about a vertical axis, what will be the locus of the apparent image of a fixed point one foot from the axis?

683. The hypotenuse of a right triangle is 10 in. long and one of the acute angles is 30°. Compute the lengths of the segments into which the short side is divided by the bisector of the opposite angle.

684. A chord BC of a given circle is drawn, and a point A moves on the longer arc BC. Draw triangle ABC and find the locus of the center of a circle inscribed in this triangle.

685. Three equal circular plates are so placed that each touches the other two, and a string is tied tightly around them. If the length of the string is 10 ft., find the radius of the circles correct to three figures.

APPENDIX

339. Contraposite Law. This law was stated in § 8, but no explanation of why it was true was attempted. The following explanation is simple, and shows very plainly that the law holds for all statements.

Given. If A, then B.

To prove. If not B, then not A.

Proof. I. If not B, then either (1) A,

or (2) not ▲ (all possibilities).

II. But, if A, then B (given).

... using (1), if not B, then A, then B. III. This is impossible, for it contradicts itself. ... If not B, then not A

(only other possibility).

(This proof depends upon the "Law of Excluded Mean," but for the purpose of this proof it is not necessary to discuss that Law.)

340. Law of Converse. Stated in § 10.

Given.

If A, then X,

If B, then Y,

If c, then Z, etc.

Where A, B, C, cover all possibilities,

...

and no two of the conclusions X, Y, Z, can be true at once.

To prove. If X, then A;

If Y, then B;

If Z, then C, etc.