137. Parts proportional to 1, 2, 3.

138. To divide a triangle into two equivalent parts by a line perpendicular to one side.

139. To divide a triangle into three equivalent parts by lines parallel to one of the medians.

140. To divide a triangle into four equivalent parts by lines parallel to one of the bisectors.

141. To divide a triangle into two equivalent parts by a line drawn through a given point.

142. To divide a triangle into two parts in the ratio 2:5 by a line drawn through a given point.

To divide by lines drawn from one vertex :

143. A parallelogram into three equivalent parts. 144. A parallelogram into four equivalent parts.

145. A parallelogram into five equivalent parts. 146. A parallelogram into two parts in the ratio 3 : 4. 147. A trapezoid into two equivalent parts. 148. A trapezoid into three equivalent parts.

149. A trapezium into two equivalent parts.

150. A trapezium into two parts in the ratio 1: 2. 151. An octagon into two equivalent parts.

To divide a parallelogram by lines drawn from a given point in one of the sides into:

152. Two equivalent parts. 153. Five equivalent parts. 154. Two parts in the ratio 3: 5.

To divide a parallelogram into two equivalent parts by a line:

155. Drawn through a given point P.

156. Parallel to a given line L.

To divide a trapezoid into two equivalent parts by a line:

157. Parallel to the bases.

158. Perpendicular to the bases.

159. Parallel to one of the legs.

160. Drawn through a given point in one of the bases. 161. Drawn through a given point P.

162. Parallel to a given line L.

163. To divide a trapezoid by lines parallel to the bases into three parts proportional to the numbers 1, 2, 3.

164. To divide a trapezium into two equivalent parts by a line drawn through a given point in one of the sides. 165. To cut from a given polygon a triangle equivalent to a given square.

To divide a hexagon by a line drawn through a given point in one of the sides into :

166. Two equivalent parts.

167. Two parts in the ratio 2: 3.

168. To inscribe in a given circle a rectangle of given area. 169. To inscribe in a given triangle a rectangle of given

area.

170. To inscribe in a given parallelogram a rhombus of given area.

CHAPTER V.

REGULAR FIGURES.

§ 22. THEOREMS.

1. The side of a circumscribed equilateral triangle is equal to twice the side of the inscribed equilateral triangle. What is the ratio of their areas?

2. The area of a circumscribed square is equal to twice the area of the inscribed square. What is the ratio of their sides?

3. The apothem of an inscribed equilateral triangle is equal to one-half the side of the inscribed regular hexagon.

4. The apothem of an inscribed regular hexagon is equal to one-half the side of the inscribed equilateral triangle.

5. An inscribed regular hexagon is twice as large as the inscribed equilateral triangle.

6. A regular inscribed hexagon is one-half as large as the circumscribed equilateral triangle.

7. The area of a regular dodecagon is equal to three times the square of its radius.

8. Upon the six sides of a regular hexagon squares are constructed outwardly. Prove that the exterior vertices of these squares are the vertices of a regular dodecagon.

9. In a regular pentagon all the diagonals are drawn. Prove that another regular pentagon is thereby formed,

10. The apothem of an inscribed regular pentagon is equal to one-half the sum of the radius of the circle and the side of the inscribed regular decagon.

11. The side of an inscribed regular pentagon is equal to the hypotenuse of a right triangle of which the legs are the radius of the circle and the side of the inscribed regular decagon.

12. The radius of an inscribed regular polygon is the mean proportional between the apothem and the radius of the similar circumscribed regular polygon.

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

14. The alternate vertices of a regular hexagon are joined by straight lines. Prove that another regular hexagon is formed. Find the ratio of the areas of the two hexagons.

15. If upon the legs of a right triangle semi-circumferences are described outwardly, the sum of the areas contained between these semi-circumferences and the semicircumference passing through the three vertices, is equal to the area of the triangle.

16. If the diameter of a circle is divided into two parts, and upon these parts semi-circumferences are described on opposite sides of the diameter, these semi-circumferences will divide the circle into two parts which have the same ratio as the two segments of the diameter.

17. If in a circle two chords are drawn perpendicular to each other, and upon the four segments of these chords as diameters circles are described, the sum of the areas of the four circles will be equal to the area of the given circie.

NOTE. When

$23. NUMERICAL EXERCISES.

occurs as a factor, take π = 24 if English units are

used, and 3.1416 if metric units are used.

=

p = apothem of inscribed regular polygon of n sides. a = one side of inscribed regular polygon of n sides. b = one side of inscribed regular polygon of 2n sides. c = one side of circumscribed regular polygon of n sides. F= area of inscribed regular polygon of n sides. G= area of inscribed regular polygon of 2n sides. In case approximate rational results are desired, √2 1.41421, √3 = 1.73205, √52.23606.

Given.

=

Required.