bisect A B, A C in F, G; and show that the triangle ABC is equal to the sum or difference of the triangles B D F, CE G, according as the angles at the base of A B C are both or only one acute.

202. If of the four triangles into which the diagonals divide a quadrilateral, two opposite ones are equal, the quadrilateral has two opposite sides parallel.

203. Upon stretching two chains A C, B D, across a field, A BCD, I find that AC, BD make equal angles with CD, and that AC makes with AD the same angle that BC does with BD: hence prove that A B is parallel to C D.

204. The two triangles formed by drawing lines from any point between two opposite sides of a parallelogram to the extremities of those sides are together half the parallelogram.

205. The difference between two triangles formed. by drawing lines from a point not between two opposite sides of a parallelogram to the extremities of those sides is equal to half the parallelogram.

206. If from the ends of one of the non-parallel sides of a trapezium two lines be drawn to the bisection of the opposite side, the triangle thus formed with the first side is half the trapezium.

207. In the figure, Euc. I., 47, show that if B G and C H be joined, these lines will be parallel.

208. In ditto, if DB, EC be produced to meet FG and K H in M, N, the triangles BF M, CK N are equiangular and equal to the triangle A B C.

HerM

209. In ditto, if G H, KE, FD be joined, each of the triangles so formed is equal to the given triangle A B C.

210. In ditto, produce FG, KH to meet in M, join M B, M C, and produce MA to cut B C in N; then show that MN is perpendicular to BC, and thence that the three lines AN, BK, CP intersect in one point.

III. PROBLEMS ON BOOK II.1

211. If a line be drawn from one of the acute angles of a right-angled triangle to the bisection of the opposite side, the square upon that line is less than the square upon the hypotenuse by three times half the line bisected.

the square upon

212. If from the middle point of one of the sides of a right-angled triangle a perpendicular be drawn to the hypotenuse, the difference of the squares of the segments so formed is equal to the square of the other side.

213. In any triangle, if a perpendicular be drawn from the vertex to the base, the difference of the squares upon the sides is equal to the difference of the squares upon the segments of the base.

These problems, as well as several in the last section of problems in Book I., are taken from the collection in Colenso's Euclid. But I have gone through all the fifty, and have made some necessary corrections. To the student who has gone carefully through the preceding pages none of these fifty problems will present any difficulty.

214. Let A OB be a quadrant of a circle, whose centre is O; from any point C in its arc draw CD perpendicular to OA or OB, meeting in E the radius which bisects the angle A OB: then show that the squares upon C D, E are together equal to the square upon O A.

215. If from any point in the diameter of a semicircle two lines be drawn to the circumference, one to the bisection of the arc, and the other perpendicular to the diameter, then the squares upon these two lines are together double of the square upon the radius.

216. If A be the vertex of an isosceles triangle ABC, and CD be drawn perpendicular to AB, prove that the squares upon the three sides are together equal to the square on BD, and twice the square on A D, and thrice the square on CD.

217. If from any point perpendiculars be dropped on all the sides of any rectilineal figure, the sum of the squares upon the alternate segments of the sides will be equal.

218. If from one of the acute angles of a rightangled triangle a line be drawn to the opposite side, the squares of that side and the line so drawn are together equal to the squares of the hypotenuse and the segment adjacent to the right angle.

219. Describe a square equal to the difference of two given squares.

220. Divide, when possible, a given line into two parts, so that the sum of their squares may be equal to a given square.

221. From D the middle point of A C, one of the sides of an equilateral triangle ABC, draw DE perpendicular on BC; and show the square upon BD is three-fourths of the square upon BC, and the line B E three-fourths of B C.

222. If from the vertex A, of a right-angled triangle BA C, AD be dropped perpendicular on the base, show that the rectangles contained by BC and BD, BC and C D, BD and CD are respectively equal to the squares upon A B, A C, A D.

223. Produce a given line so that the rectangle of the whole line produced and the original line shall be equal to a given square.

224. If on the radius of a circle a semicircle be described, and a perpendicular to the common diameter be drawn, the square of the chord of the greater circle, between the extremity of the diameter and the point of section of the perpendicular, will be double of the square of the corresponding chord of the lesser circle.

225. Divide a line in two points equally distant from its extremities, so that the square on the middle part shall be equal to the sum of the squares on the extremes; and show also that in this case the square of the whole line will be equal to the squares of the extreme parts together with twice the rectangle of the whole and the middle part.

226. Divide a line into two parts, so that the squares of the whole line and one of the parts shall be together double of the square of the other part:

and show that, by the same division, the square of the greater part will be equal to twice the rectangle of the whole and the lesser part.

227. Divide a straight line into two parts so that the sum of their squares may be the least possible.

228. Show that the sum of the squares upon two lines is never less than twice their rectangle, and that the difference of their squares is equal to the rectangle of their sum and difference.

229. Show that of the two algebraical expressions, (a+x) (a−x) + x2=a2, (a+x)2 + (a−x)2=2a2 + 2x2, the first is equivalent to Props. 5 and 6, and the second to Props. 9 and 10, of Euc. II.

230. A B C D is a rectangle, E any point in BC, F in CD: show that the rectangle A B C D is equal to twice the triangle A EF together with the rectangle BE, DF.

231. If a line be divided into two equal and also into two unequal parts, the squares of the two unequal parts are together equal to twice the rectangle contained by these parts together with four times the square of the line between the points of section.

232. If from one of the equal angles of an isosceles triangle a perpendicular be dropped on the opposite side, the rectangle of that side and the segment of it between the perpendicular and base is equal to half the square upon the base.

233. A, B, C, D, are four points in the same line; E a point in that line equally distant from the middle of the segments A B, CD; F any other point