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 DA; 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 O be the centre of a circle and OA a radius and a circle be described on OA as diameter, the circum-
