Show that the triangle B F E is equal in all respects. to the triangle ABC.

Show that the same holds good if BFC lies towards the same parts as the parallelogram and A B E, C D G towards opposite parts.

173. In the parallelogram ABCD, the angle A D B is equal to the angle AC B. Show that ABCD is rectangular.

174. In the parallelogram ABCD, the angle ADB is equal to one-third part of the angle AEB; also A C and B D intersect at an angle equal to onethird part of two right angles. Show that one of the diagonals is at right angles to opposite sides of the parallelogram.

175. If the angle between two adjacent sides of a parallelogram be increased, while their lengths remain unchanged, the diagonal through the point of intersection will be diminished.

176. If two opposite sides of a parallelogram be bisected, the lines drawn from the points of bisection to the opposite sides trisect the diagonal.

177. If AB, a side of the parallelogram ABCD, be divided into n equal parts, show that a line drawn from C to the division point nearest to B cuts off from the diagonal B D one (n + 1)th part,―measured from B.

Take for n any convenient number-say 7. Divide A B and C D into 7 equal parts, and join C with the division nearest to B, the division nearest to C with the next division from B, and so on. It will then be easy,

in the manner of the preceding example, to show that any one of the 8 parts into which the diagonal is thus divided is equal to any other part-or, in other words, that the diagonal is divided into 8 equal parts.

178. In the straight line A B C, A B is equal to BC. Show that perpendiculars drawn from the points A, B, and C upon any straight line meet it in equidistant points.

(i) When the line passes between A and C.

(ii) When the line does not pass between A and C.

179. In case (ii) of Example 178, show that the perpendicular from A and C are together double of the perpendicular from B.

180. In case (i) of Example 178, show that the difference of the perpendiculars from A and C is double of the perpendicular from B.

181. If straight lines be drawn from the angles of any parallelogram perpendicular to a straight line which is outside the parallelogram, the sum of those from one pair of opposite angles is equal to the sum of those from the other pair of opposite angles.

182. Determine a point in the base of a triangle from which lines drawn parallel to the sides, to meet them, are equal.

183. If an hexagonal figure admits of division into three parallelograms, each pair of opposite sides are equal and parallel.

Show that in general such an hexagonal figure

admits of being divided into three parallelograms in two different ways.

184. If each pair of opposite sides of a hexagon are parallel, and one pair equal, the other pairs are also equal.

185. If each pair of opposite sides of a hexagon be equal and parallel, the three straight lines joining opposite angles will meet in a point.

186. If each pair of opposite sides of a rectilinear figure having an even number of sides be equal and parallel, all the lines joining opposite angles meet in a point.

187. Describe a rhombus within a given parallelogram, so that one of the angular points may occupy a given point on the perimeter of the parallelogram.

188. Describe a rectangle within a given parallelogram, so that one of the angular points may occupy a given point on the perimeter of the parallelogram.

In Examples 187 and 188 it suffices that the angles of the constructed figures should lie on the sides or the sides produced of the parallelogram. Previous examples show the relations which hold when a parallelogram is a rhombus or rectangular, and these will be found sufficient for the solution of Examples 187 and 188.

189. The three sides of a triangle are together less than the three lines drawn from the angles to the bisections of the opposite sides.

Complete a parallelogram having two sides of the triangle as adjacent sides. Then show that these sides

are together greater than the diagonal which passes through the bisection of the base, &c.

II. PROBLEMS ON PROPOSITIONS 35 TO THE
END OF BOOK I.

190. On the sides A B, A C of a triangle describe parallelograms ABDE, ACFG, and produce DE, FG to meet in H: then the area of these parallelograms together is equal to the area of the parallelogram on BC, whose side is equal and parallel to A H.

Draw this parallelogram, and show that HA produced divides it into parts equal respectively to AD and AF.

191. From a given point in one of the equal sides of an isosceles triangle draw a line, meeting the other side produced, which shall make with these sides a triangle equal to the given triangle. Let A B, A C be the equal sides; F the given point in A C; and let C D perpendicular to BC meet B A produced in D; draw CE parallel to FD, cutting BD produced in E; then F E A is the required triangle.

192. If one angle of a triangle be a right angle, and another be two-thirds of a right angle, show that the equilateral triangle on the hypothenuse is equal in area to the sum of those on the sides.

193. Convert a trapezium into a triangle of equal area with one angle common.

194. Given a triangle ABC and a point D in

A B; construct another triangle A D E equal to the former, and having the common angle A.

195. Change a triangle into another equal one of given altitude.

196. If the sides of any quadrilateral be bisected and the points of bisection joined, the included figure is a parallelogram, and equal in area to half the original figure; show also that the lines joining the bisections of opposite sides bisect each other.

There is a pretty statical proof of the last property resulting from the determination of the centre of gravity of four equal particles at A, B, C, and D.

197. Through D, E, the bisections of the sides A B, AC of a triangle, draw DF, EF parallel to BE, AB; and show that the sides of the triangle DCF are equal to the three lines drawn from the angles to bisect the sides.

198. Bisect a triangle by a line drawn from a given point in one of its sides.

199. If from any point in the diagonal of a parallelogram lines be drawn to the angles, the parallelogram will be divided into two pairs of equal triangles.

200. Through E, the bisection of the diagonal BD of a quadrilateral A B C D, draw F E G parallel to A C; and show that A G will bisect the figure.

201. A B C is a given triangle; draw BD, CE perpendicular to BC and on the same side of it, each equal to twice the altitude of the triangle;