107. A hexagon has its three pairs of opposite sides parallel. Prove that the two triangles which can be formed by joining alternate vertices are of equal area.

108. A quadrilateral and a triangle are such that two of the sides of the triangle are equal to the two diagonals of the quadrilateral and the angle between these sides is equal to the angle between the diagonals. Prove the areas of the quadrilateral and triangle are equal.

109. Prove that the straight lines drawn from the corners of a square to the middle points of the opposite sides taken in order form a square of one-fifth the area of the original square.

110. The area of the octagon formed by the straight lines joining each vertex of a parallelogram to the middle points of the two opposite sides is one-sixth the area of the parallelogram.

111. ABCD is a parallelogram. A point E is taken on CD such that CE is an nth part of CD; the diagonal AC cuts BE in F. Prove the following continued proportion connecting the areas of the parts of the parallelogram

ADEFA: AFB : BFC : CFE=n2+n-1:n2:n:I

112. The squares ACKE and BCID are constructed on the sides of a right triangle ABC; the lines AD and BE intersect at G; AD cuts CB in H, and BE cuts AC in F. Prove that the quadrilateral FCHG and the triangle ABG are equivalent.

PROBLEMS OF CONSTRUCTION

113. Construct an equilateral triangle which shall be equal in area to a given parallelogram.

114. Construct a square which shall have a given ratio to a given

square.

115. A pavement is made of black and white tiles, the black being squares, the white equilateral triangles whose sides are equal to the sides of the squares. Construct the pattern so that the areas of black and white may be in the ratio √3:4.

116. Produce a given straight line so that the square on the whole line shall have a given ratio to the rectangle contained by the given line and its extension. When is the problem impossible?

117. Find a point in the base produced of a triangle such that a straight line drawn through it cutting a given area from the triangle may be divided by the sides of the triangle into segments having a given ratio.

118. Bisect a given quadrilateral by a straight line drawn through

a vertex.

PROBLEMS FOR COMPUTATION

119. (1.) If the area of an equilateral triangle is 164.51 sq. in., find its perimeter.

(2.) The perimeter of an equilateral pentagon is 25.135 ft. Its area is 23.624 sq. ft. Find the area of a similar pentagon one of whose sides is 10.361 ft.

(3.) Find, in acres, the area of a triangle, if two of its sides are 16.342 rds. and 23.461 rds., and the included angle is 135°.

(4.) Find the area of the triangle in the preceding example in hec

tares.

(5.) The sides of a triangle are 13.461, 16.243, and 20.042 miles. Find the areas of the parts into which it is divided by any median.

(6.) The sides of a triangle are 12 in., 15 in., and 17 in. Find the areas of the parts into which it is divided by the bisector of the smallest angle.

(7.) Two sides of a triangle are in the ratio 2 to 5. Find the ratio of the parts into which the bisector of the included angle divides the triangle.

(8.) The altitude upon the hypotenuse of a right triangle is 98.423 in. One part into which the altitude divides the hypotenuse is four times the other. Find the area of the triangle.

(9.) Find the perimeter of the triangle in the preceding example. (10.) The areas of two similar polygons are 22.462 sq. in. and 14.391 sq. m. A side of the first is 2 in. Find the homologous side of the second.

(11.) The sides of a triangle are .016256, .013961, and .020202. Find the radius of the inscribed circle.

(12.) A mirror measuring 33 in. by 22 in. is to have a frame of uni

form width whose area is to equal the area of the mirror; find what the width of the frame should be.

(13.) The sum of the radii of the inscribed, circumscribed, and an escribed circle of an equilateral triangle is unity. What is the area of the triangle ?

BOOK V

PROBLEMS OF DEMONSTRATION

120. An equilateral polygon inscribed in a circle is regular. An equilateral polygon circumscribed about a circle is regular, if the number of sides is odd.

121. An equiangular polygon inscribed in a circle is regular if the number of sides is odd. An equiangular polygon circumscribed about a circle is regular.

122. The diagonals of a regular pentagon are equal.

123. The pentagon formed by the diagonals of a regular pentagon is regular.

124. An inscribed regular octagon is equivalent to a rectangle whose sides are equal to the sides of an inscribed and a circumscribed square.

125. If a triangle is formed having as sides the radius of a circle, the side of an inscribed regular pentagon, and the side of an inscribed regular decagon, this triangle will be a right triangle.

126. The area of a regular hexagon inscribed in a circle is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles.

127. If perpendiculars are drawn from the vertices of a regular polygon to any straight line through its centre, the sum of those which fall upon one side of the line is equal to the sum of those which fall upon the other side.

128. The area of any regular polygon inscribed in a circle is a mean proportional between the areas of the inscribed and circumscribed polygons of half the number of sides.

129. If, on the sides of a right triangle as diameters, semi-circumferences are described exterior to the triangle, and a circumference is drawn through the three vertices, the sum of the crescents thus formed is equivalent to the triangle.

130. If two circles are internally tangent to a third circle and the sum of their radii is equal to the radius of the third circle, the shorter arc of the third circle comprised between their points of contact is equal to the sum of the arcs of the two small circ'es from their points of contact with the third circle to their intersection which is nearest the large circle.

131. If CD is the perpendicular from the vertex of the right angle of a right triangle ABC, prove that the areas of the circles inscribed in the triangles ACD, BCD are proportional to the areas of the triangles.

PROBLEMS OF CONSTRUCTION

132. To construct a circumference whose length shall equal the sum of the lengths of two given circumferences.

133. To construct a circle equivalent to the sum of two given circles. 134. To inscribe a regu ar octagon in a given square. 135. To inscribe a regular hexagon in a given equilateral triangle. 136. Divide a given circle into any number of parts proportional to given straight lines by circumferences concentric with it.

137. Find four circles whose radii are proportional to given lines, and the sum of whose areas is equal to the area of a given circle. 138. In a given equilateral triangle inscribe three equal circles each tangent to the two others and to two sides of the triangle.

139. In a given circle inscribe three equal circles each tangent to the two others and to the given circle.

140. The length of the circumference of a circle being represented by a given straight line, find approximately by a geometrical construction the radius.

PROBLEMS FOR COMPUTATION

141. (1.) A regular octagon is inscribed in a circle whose radius is 4 ft. Find the segment of the circle contained between one side of the octagon and its subtended arc.

(2.) Find the area of an equilateral triangle circumscribed about a circle whose radius is 14.361 in.

(3.) An isosceles right triangle is circumscribed about a circle whose radius is 3 cm. Find (a) each side; (b) its area; (c) the area in each corner of the triangle bounded by the circumference of the circle and two sides of the triangle.

(4.) Find the area of the circle inscribed in an equilateral triangle, one side of which is 7.4631 ft.

(5.) Find the difference between the area of a triangle whose sides are 4.6213 mm., 3.7962 mm., and 2.6435 mm., and the area of the circumscribed circle.

(6.) The area of a circle is 14632 sq. ft. Find its circumference in yards.

(7.) Find the area of a ring whose outer circumference is 15.437 ft., and whose inner circumference is 9.3421 ft.

(8.) Find the ratio of the areas of two circles inscribed in equilateral triangles, if the perimeter of one triangle is four times that of the other.

(9.) If the area of an equilateral triangle inscribed in a circle is 12 sq. ft., what is the area of a regular hexagon circumscribed about the same circle?

(10.) Find the side of a regular octagon whose area shall equal the sum of the areas of two regular hexagons, one inscribed in and the other circumscribed about a circle whose radius is 10.462 in.

(11.) A man has a circular farm 640 acres in extent. He gives to each of his four sons one of the four largest equal circular farms which can be cut off from the original farm. How much did each son receive?

(12.) A man has a circular tract of land 700 acres in area; he wills one of the three largest equal circular tracts to each of his three sons, the tract at the centre included between the three circular tracts to his daughter, and the tracts included between the circumference of the original tract and the three circular tracts to his wife. How much will each receive?

(13.) A man owned a tract of land 323,250 sq. m. in area, and in the