208. ABCD is a quadrilateral inscribed in a circle, and the sides AB, DC when produced meet at O: shew that the triangle AOC is equiangular to the triangle BOD.

209. Shew that no parallelogram except a rectangle can be inscribed in a circle.

210. A triangle is inscribed in a circle: shew that the sum of the angles in the three segments exterior to the triangle is equal to four right angles.

211. A quadrilateral is inscribed in a circle: shew that the sum of the angles in the four segments of the circle exterior to the quadrilateral is equal to six right angles.

212. Divide a circle into two parts so that the angle contained in one segment shall be equal to twice the angle contained in the other.

213. Divide a circle into two parts so that the angle contained in one segment shall be equal to five times the angle contained in the other.

214. If the angle contained by any side of a quadrilateral and the adjacent side produced, be equal to the opposite angle of the quadrilateral, shew that any side of the quadrilateral will subtend equal angles at the opposite angles of the quadrilateral.

215. If any two consecutive sides of a hexagon inscribed in a circle be respectively parallel to their opposite sides, the remaining sides are parallel to each other.

216. A, B, C, D are four points taken in order on the circumference of a circle; the straight lines AB, DC produced intersect at P, and AD, BC at Q: shew that the straight lines which respectively bisect the angles APC, AQC are perpendicular to each other.

217. If a quadrilateral be inscribed in a circle, and a straight line be drawn making equal angles with one pair of opposite sides, it will make equal angles with the other pair.

218. A quadrilateral can have one circle inscribed in it and another circumscribed about it: shew that the straight lines joining the opposite points of contact of the inscribed circle are perpendicular to each other.

III. 23 to 30.

219. The straight lines joining the extremities of the chords of two equal arcs of a circle, towards the same parts are parallel to each other.

220. The straight lines in a circle which join the extremities of two parallel chords are equal to each other.

221. AB is a common chord of two circles; through C any point of one circumference straight lines CAD, ČBE are drawn terminated by the other circumference: shew that the arc DE is invariable.

222. Through a point in the circumference of a circle two straight lines ACB, DCE are drawn cutting the circle at B and E: shew that the straight line which bisects the angles ACE, DCB meets the circle at a point equidistant from B and E.

223. The straight lines bisecting any angle of a quadrilateral inscribed in a circle and the opposite exterior angle, meet in the circumference of the circle.

224. AB is a diameter of a circle, and D is a given point on the circumference, such that the arc DB is less than half the arc DA: draw a chord DE on one side of AB so that the arc EA may be three times the arc BD.

225. From A and B two of the angular points of a triangle ABC, straight lines are drawn so as to meet the opposite sides at P and Q in given equal angles: shew that the straight line joining P and Q will be of the same length in all triangles on the same base AB, and having vertical angles equal to C.

226. If two equal circles cut each other, and if through one of the points of intersection a straight line be drawn terminated by the circles, the straight lines joining its extremities with the other point of intersection are equal.

227. OA, OB, OC are three chords of a circle; the angle AOB is equal to the angle BOC, and OA is nearer to the centre than OB. From B a perpendicular is drawn on OA, meeting it at P, and a perpendicular on OC produced, meeting it at Q: shew that AP is equal to CQ.

228. AB is a given finite straight line; through A two indefinite straight lines are drawn equally inclined to AB; any circle passing through A and B meets these straight lines at L and M. Shew that if AB be between AL and AM the sum of AL and AM is constant; if AB be not between AL and AM the difference of AL and AM is constant.

229. AOB and COD are diameters of a circle at right angles to each other; E is a point in the arc AC, and EFG is a chord meeting COD at F, and drawn in such a

direction that EF is equal to the radius. Shew that the arc BG is equal to three times the arc AE.

230. The straight lines which bisect the vertical angles of all triangles on the same base and on the same side of it, and having equal vertical angles, all intersect at the same point.

231. If two circles touch each other internally, any chord of the greater circle which touches the less shall be divided at the point of its contact into segments which subtend equal angles at the point of contact of the two circles.

III. 31.

232. Right-angled triangles are described on the same hypotenuse: shew that the angular points opposite the hypotenuse all lie on a circle described on the hypotenuse as diameter.

233. The circles described on the equal sides of an isosceles triangle as diameters, will intersect at the middle point of the base.

234. The greatest rectangle which can be inscribed in a circle is a square.

235. The hypotenuse AB of a right-angled triangle ABC is bisected at D, and EDF is drawn at right angles to AB, and DE and DF are cut off each equal to ĎA; CE and CF are joined: shew that the last two straight lines will bisect the angle C and its supplement respectively.

236. On the side AB of any triangle ABC as diameter a circle is described; EF is a diameter parallel to BC: shew that the straight lines EB and FB bisect the interior and exterior angles at B.

237. If AD, CE be drawn perpendicular to the sides BC, AB of a triangle ABC, and DE be joined, shew that the angles ADE and ACE are equal to each other.

238. If two circles ABC, ABD intersect at A and B, and AC, AD be two diameters, shew that the straight line CD will pass through B.

239. If be the centre of a circle and OA a radius and a circle be described on OA as diameter, the circum

ference of this circle will bisect any chord drawn through it from A to meet the exterior circle.

240. Describe a circle touching a given straight line at a given point, such that the tangents drawn to it from two given points in the straight line may be parallel.

241. Describe a circle with a given radius touching a given straight line, such that the tangents drawn to it from two given points in the straight line may be parallel.

242. If from the angles at the base of any triangle perpendiculars are drawn to the opposite sides, produced if necessary, the straight line joining the points of intersection will be bisected by a perpendicular drawn to it from the centre of the base.

243. AD is a diameter of a circle; B and Care points on the circumference on the same side of AD; a perpendicular from D on BC produced through C, meets it at E: shew that the square on AD is greater than the sum of the squares on AB, BC, CD, by twice the rectangle BC, CE.

244. AB is the diameter of a semicircle, P is a point on the circumference, PM is perpendicular to AB; on AM, BM as diameters two semicircles are described, and AP, BP meet these latter circumferences at Q, R: shew that QR will be a common tangent to them.

245. AB, AC are two straight lines, B and C are given points in the same; BD is drawn perpendicular to AC, and DE perpendicular to AB; in like manner CF is drawn perpendicular to AB, and FG to AC. Shew that EG is parallel to BC.

246. Two circles intersect at the points A and B, from which are drawn chords to a point Cin one of the circumferences, and these chords, produced if necessary, cut the other circumference at D and E: shew that the straight line DE cuts at right angles that diameter of the circle ABC which passes through C.

247. If squares be described on the sides and hypotenuse of a right-angled triangle, the straight line joining the intersection of the diagonals of the latter square with the right angle is perpendicular to the straight line joining the intersections of the diagonals of the two former.

248. C is the centre of a given circle, CA a straight line less than the radius; find the point of the circumference at which CA subtends the greatest angle.

249. AB is the diameter of a semicircle, D and E are any two points in its circumference. Shew that if the chords joining A and B with D and E each way intersect at F and G, then FG produced is at right angles to AB.

250. Two equal circles touch one another externally, and through the point of contact chords are drawn, one to each circle, at right angles to each other: shew that the straight line joining the other extremities of these chords is equal and parallel to the straight line joining the centres of the circles.

251. A circle is described on the shorter diagonal of a rhombus as a diameter, and cuts the sides; and the points of intersection are joined crosswise with the extremities of that diagonal: shew that the parallelogram thus formed is a rhombus with angles equal to those of the first.

252. If two chords of a circle meet at a right angle within or without a circle, the squares on their segments are together equal to the squares on the diameter.

III. 32 to 34.

253. B is a point in the circumference of a circle, whose centre is C; PA, a tangent at any point P, meets CB produced at A, and PD is drawn perpendicular to CB: shew that the straight line PB bisects the angle APD.

254. If two circles touch each other, any straight line drawn through the point of contact will cut off similar segments.

255. AB is any chord, and AD is a tangent to a circle at A. DPQ is any straight line parallel to AB, meeting the circumference at P and Q. Shew that the triangle PAD is equiangular to the triangle QAB.

256. Two circles ABDH, ABG, intersect each other at the points A, B; from B a straight line BD is drawn in the one to touch the other; and from A any chord whatever is drawn cutting the circles at G and H: shew that BG is parallel to DÍ.

257. Two circles intersect at A and B. At A the tangents AC, AD are drawn to each circle and terminated