10. Given any triangle, to construct an equivalent isosceles triangle
having the base equal to a given length.

11. Given a square, to construct an equivalent equilateral triangle.
See Ex. 1.

12. Given any triangle, to construct an equivalent equilateral tri-
angle.

13. Given a square, to construct an equivalent right triangle having
one leg equal to a given line.

14. Given a square, to construct an equivalent rectangle having one
side equal to a given line.

15. Given a parallelogram, to construct an equivalent rectangle having
a given altitude.

16. Given a parallelogram, to construct an equivalent parallelogram
having one side equal to a given line.

17. Given a parallelogram, to construct an equivalent parallelogram
having one angle equal to a given angle.

18. To construct a circle which shall pass through two given points
and be tangent to a given line.

19. To construct a circle which shall pass through a given point and be
tangent to two straight lines.

20. To construct a line, having given the greater segment of the line
divided into extreme and mean ratio.

21. To construct two straight lines, having given their sum and their
difference.

22. To construct two straight lines, having given their sum and their
ratio.

23. To construct two straight lines, having given their difference and
their ratio.

24. To divide a triangle into two equivalent parts by drawing a line
through a given point on one of its sides.

25. To divide a triangle into two equivalent parts by drawing a line
parallel to one of its sides.

26. To divide a parallelogram into two equivalent parts by drawing a
line through a given point in one of its sides.

27. To divide a trapezoid into two equivalent parts by drawing a line
parallel to its bases.

28. To divide a trapezoid into two equivalent parts by drawing a line
through a given point in one of its bases.

29. To construct a trapezoid with a given lower base equivalent to a
given trapezoid.

THEOREMS IN BOOK V.

1. Every equilateral polygon circumscribed about a circle is regular if the number of its sides is odd.

2. Every equiangular polygon inscribed in a circle is regular if the number of its sides is odd.

3. The area of the inscribed regular dodecagon is equal to three times the square of the radius.

4. The diagonals of a regular pentagon joining the alternate vertices form a regular pentagon.

5. The diagonals joining the alternate vertices of a regular hexagon form a regular hexagon.

6. The regular hexagon formed by joining the alternate vertices of a regular hexagon equals one-third of the given hexagon.

7. If squares are described outwardly on the sides of a regular hexagon, their exterior vertices are the vertices of a regular dodecagon.

8. The area of an inscribed regular octagon is equal to the area of a rectangle whose adjacent sides are equal to the sides of the inscribed and the circumscribed squares.

9. A plane surface may be entirely covered (as in the pavement of a street) by equal regular polygons of either three, four, or six sides.

10. The area of a circular ring is equal to the area of a circle whose diameter is a chord of the outer circle and a tangent of the inner circle.

11. The square of the side of an inscribed regular pentagon is equal to the sum of the squares of the radius of the circle and the side of the inscribed regular decagon.

12. If a right triangle has an inscribed and a circumscribed circle, the sum of their diameters will be equal to the sum of the sides containing the right angle.

13. Two diagonals of a regular pentagon, not drawn from a common vertex, divide each other in extreme and mean ratio.

14. If semicircles are described on the legs of a right triangle as diameters, and from the whole figure a semicircle on the hypotenuse is subtracted, the remainder is equivalent to the triangle.

15. A plane surface may be entirely covered by a combination of squares and regular octagons having the same side, or by equilateral triangles and dodecagons having the same side.

16. The side of an inscribed equilateral triangle is equal to the hypotenuse of a right triangle whose sides are the sides of the inscribed square and the inscribed regular hexagon, respectively.

PROBLEMS IN BOOK V.

1. DIVIDE a given circle into any given number of equivalent parts by concentric circumferences.

2. In a given equilateral triangle, inscribe three equal circles tangent to one another and to the sides of the triangle.

SUGGESTION.-Find the radius of the circles in terms of the sides of the

triangle.

3. In a given circle, inscribe three equal circles tangent to one another and to the given circle.

SUGGESTION.

the given circle.

Find the radius of the circles in terms of the radius of

4. Given three equal circles tangent to one another, to circumscribe a circle about them.

DERIVED GENERAL EXPRESSIONS.

1. Find the altitude of an equilateral triangle whose side is denoted by a.

SOLUTION.—In ▲ ABC, AB=a, DB=2.

2. Find the side of an equilateral triangle in terms of its altitude

3. Find the area, S, of an equilateral triangle in terms of its side

== a.

h.

_ah=h; but h=3. (Ex. 1).

Hence,

a

=

4

4. Find the area of an equilateral triangle in terms of its altitude = h.

5. Find the base of an isosceles triangle in terms of its altitude = h, and a leg = a. Ans. Base = 2(a2 — h3)ž.

6. Find the altitude of an isosceles triangle in terms of its base = b, and Ans. Alt.√ 4a2 — b2.

a leg = a,

7. Find the area of an isosceles triangle in terms of its base = b, and a leg = a.

8. Find the area of a right triangle if one leg = a and the altitude upon the hypotenuse = h.

SOLUTION. In the ▲ ABC, right angled at C,

9. Find the area of a triangle whose sides are a, b, and c, in terms of

the radius =r of the inscribed circle.

Ans. S = (a+b+c).

10. Find the area of a triangle in terms of the radius of the circumscribing circle.

SOLUTION.-Let R denote the radius of the cir

11. Find the medians of a triangle in terms of its side.

SOLUTION.-Let a, b, c denote the sides, and m the median. Then, by

12. What is the area of an equilateral triangle whose side is twelve inches?

13. What is the area of an isosceles triangle whose base is twelve inches and one leg sixteen inches?

14. Find the altitude of a triangle in terms of

its sides.

SOLUTION.-Denote the sides and altitude as in

the figure.

a

B

b

15. Find the area of a triangle in terms of its sides.

SOLUTION.-Denote the area by A; then in the above triangle (B. IV.,

COR.-To find the area of a ▲ when its sides are given, take half the sum of the sides, subtract from it each of the sides separately, take the product of the half sum and these remainders, and extract the square root of this product.