EXERCISES.

BOOK I.

120. Show how to make a rhombus having one of its diagonals equal to a given sect.

121. If two quadrilaterals have three consecutive sides, and the two contained angles in the one respectively equal to three consecutive sides and the two contained angles in the other, the quadrilaterals are congru

ent.

122. Two circles cannot cut one another at two points on the same side of the line joining their centers.

123. Prove, by an equilateral triangle, that, if a right-angled triangle have one of the acute angles double of the other, the hypothenuse is double of the side opposite the least angle.

124. Draw a perpendicular to a sect at one extremity.

125. Draw three figures to show that an exterior angle of a trianglemay be greater than, equal to, or less than, the interior adjacent angle. 126. Any two exterior angles of a triangle are together greater than a straight angle.

127. The perpendicular from any vertex of an acute-angled triangle on the opposite side falls within the triangle.

128. The perpendicular from either of the acute angles of an obtuseangled triangle on the opposite side falls outside the triangle.

129. The semi-perimeter of a triangle is greater than any one side, and less than any two sides.

130. The perimeter of a quadrilateral is greater than the sum, and less than twice the sum, of the diagonals.

131. If a triangle and a quadrilateral stand on the same base, and the one figure fall within the other, that which has the greater surface shall have the greater perimeter.

132. If one angle of a triangle be equal to the sum of the other two, the triangle can be divided into two isosceles triangles.

133. If any sect joining two parallels be bisected, this point will bisect any other sect drawn through it and terminated by the parallels.

134. Through a given point draw a line such that the part intercepted between two given parallels may equal a given sect.

135. The medial from vertex to base of a triangle bisects the intercept on every parallel to the base.

136. Show that the surface of a quadrilateral equals the surface of a triangle which has two of its sides equal to the diagonals of the quadrilateral, and the included angle equal to either of the angles at which the diagonals intersect.

137. Describe a square, having given a diagonal.

138. ABC is a right-angled triangle; BCED is the square on the hypothenuse; ACKH and ABFG are the squares on the other sides. Find the center of the square ABFG (which may be done by drawing the two diagonals), and through it draw two lines, one parallel to BC, and the other perpendicular to BC. This divides the square ABFG into four congruent quadrilaterals. Through each mid-point of the sides of the square BCED draw a parallel to AB or AC. If each be extended until it meets the second of the other pair, they will cut the square BCED into a square and four quadrilaterals congruent to ACKH and the four quadrilaterals in ABFG.

139. The orthocenter, the centroid, and the circumcenter of a triangle are collinear, and the sect between the first two is double of the sect between the last two.

140. The perpendicular from the circumcenter to any side of a triangle is half the sect from the opposite vertex to the ortho

center.

141. Sects drawn from a given point to a given circle are bisected; find the locus of their mid-points.

142. The intersection of the lines joining the mid-points of opposite sides of a quadrilateral is the mid-point of the sect joining the mid-points of the diagonals.

143. A parallelogram has central symmetry.

SYMMETRY.

144. No triangle can have a center of symmetry, and every axis of symmetry is a medial.

145. Of two sides of a triangle, that is the greater which is cut by the perpendicular bisector of the third side.

146. If a right-angled triangle is symmetrical, the axis bisects the right angle.

147. An angle in a triangle will be acute, right, or obtuse, according as the medial through its vertex is greater than, equal to, or less than, half the opposite side.

148. If a quadrilateral has axial symmetry, the number of vertices not on the axis must be even; if none, it is a symmetrical trapezoid; if two, it is a kite.

149. A kite has the following seven properties; from each prove all the others by proving that a quadrilateral possessing it is a kite.

(1) One diagonal, the axis, is the perpendicular bisector of the other, which will be called the transverse axis.

(2) The axis bisects the angles at the vertices which it joins. (3) The angles at the end-points of the transverse axis are equal, and equally divided by the latter.

(4) Adjacent sides which meet on the axis are equal.

(5) The axis divides the kite into two triangles which are congruent, with equal sides adjacent.

(6) The transverse axis divides the kite into two triangles, each of which is symmetrical.

(7) The lines joining the mid-points of opposite sides meet on the axis, and are equally inclined to it.

150. A symmetrical trapezoid has the following five properties; from each prove all the others by proving that a quadrilateral possessing it is a symmetrical trapezoid.

(1) Two opposite sides are parallel, and have a common perpendicular bisector.

(2) The other two opposite sides are equal, and equally inclined to either of the other sides.

(3) Each angle is equal to one, and supplemental to the other, of its two adjacent angles.

(4) The diagonals are equal, and divide each other equally.

(5) One median line bisects the angle between those sides produced which it does not bisect, and likewise bisects the angle between the two diagonals.

151. Prove the properties of the parallelogram from its central symmetry.

152. A kite with a center is a rhombus; a symmetrical trapezoid with a center is a rectangle; if both a rhombus and a rectangle, it is a square.

BOOK II.

153. The perpendicular from the centroid to a line outside the triangle equals one-third the sum of the perpendiculars to that line from the vertices.

154. If two sects be each divided internally into any number of parts, the rectangle contained by the two sects is equivalent to the sum of the rectangles contained by all the parts of the one, taken separately, with all the parts of the other.

155. The square on the sum of two sects is equivalent to the sum of the two rectangles contained by the sum and each of the sects.

156. The square on any sect is equivalent to four times the square on half the sect.

157. The rectangle contained by two internal segments of a sect grows less as the point of section moves from the mid-point.

158. The sum of the squares on the two segments of a sect is least when they are equal.

159. If the hypothenuse of an isosceles right-angled triangle be divided into internal or external segments, the sum of their squares is