3. Three lines are 5, 3, and 7: find their fourth proportional.

4. Two lines are 11 and 9; find their third proportional.

5. Two lines are 10% and 4; construct their mean proportional.

6. Given the proportions, 3: 8 :: 5 : x, and 8 : 3 :: 5 : x, to construct x. 7. Given x= 17, to construct x.

Suggestion. x =

8. To divide a line 10 inches long internally in extreme and mean ratio. (See Prob. VII.)

9. To construct a triangle whose sides are 7, 9, and 11; then to construct a similar triangle whose side homologous to side 9, is 5. (See Prob. X.) 10. Through the point P, in the angle A, to draw

A

Р

a line, so that the segments between the point and the sides of the angle shall be equal. (See Prob. IX.) 11. To construct a triangle equal to a given hexa

gon.

12. The area of a square is 16; to construct a square that shall be to it in the ratio of 5 to 3. (See Prob. XVII.)

13. To construct a square that shall be equal to a triangle whose area is 32, and base 8.

14. To construct a square that shall be equal to a given pentagon plus a given parallelogram.

Suggestion. Find squares separately equal to the pentagon and parallelogram. (See Probs. XI, XII, and XIII.)

15. The areas of two squares are 25 and 16; to construct the square equal to their difference. (See Prob. XIV.)

16. The area of a rectangle is 48, and the base 8; to construct on the base 121⁄2 an equal rectangle.

r

h

17. A pentagon being given, to construct a similar pentagon whose sides shall be to those of the given pentagon in the ratio of 3 to 5.

Suggestion.-Observe the order of the terms. (See Prob. XXI.)

18. The radius of a circle is to the side of the inscribed square as 1 is to the square root of 2.

19. To construct an isosceles triangle having

either of the angles at the base twice the third angle.

Suggestion.-Draw a line, and divide it in extreme and mean ratio. (See Prob. XXIV.)

20. A hexagon being given, to construct a similar one whose area shall be in the ratio of 5 to 3 to that of the given one.

21. The area of a square is 16; to find

the length of a side of the regular inscribed octagon.

22. Given two pentagons, to construct a square equal to their difference.

23. To inscribe in a circle an equilateral triangle.

Suggestion.-Draw a diameter, and from extremity of it as centre, with radius of given circle, describe an arc cutting given circumference. Join points of intersection with the other extremity of the diameter. Prove the triangle to be equiangular.

24. To inscribe a square in a given rhombus.

Suggestion.-A B and C D are diagonals; ab and cd, bisectors of angles. How do the diagonals of a rhombus intersect each other?

a

B

Suggestion.-General method same as in the preceding. Divide the radius in extreme and mean ratio, and thence divide the semicircumference into five equal parts. How many degrees in an interior angle of a regular pentagon?

27. To construct a five-point star. Suggestion.-Divide the circumference of circle into five equal parts.

pass a circumference through A, B, and C Ch. IV, Th. IX. Let A OMB be revolved about OM (folded over)..

Since the angles at M are right, and MB = MC, B will coincide with C.

Since the angle B = the angle C, and A B = CD, A will coincide with D.

Hence, the circumference which passes through A, B, and C, will pass through D.

Similarly, the circumference passing through B, C, and D, will pass through E, and thus through all the vertices of the polygon; and hence is circumscribed. Q. E. D.

167

F

THEOREM II.

A circumference may be inscribed in any regular polygon.

M

B

E

distant from the centre

Hence, the circumference described from centre O, with radius = O M, will touch all the sides of the polygon, and is therefore inscribed.

Q. E. D.

DEFINITIONS.

From Theorems I and II we learn, that within a regular polygon there is a point equally distant from all the vertices,- the centre of the circumscribed circle; also, that the same point is equally distant from the sides, the centre of the inscribed circle; and that this point is hence the common centre of the circumscribed and inscribed circles. These considerations make clear the following definitions :

1. The Centre of a regular polygon is the common centre of the circumscribed and inscribed circles.

2. The Radius of a regular polygon is the radius of the circumscribed circle.

3. The Apothem of a regular polygon is the radius of the inscribed circle.

4. A Central Angle of a regular polygon is an angle included between two radii drawn to the extremities of the same side.