Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/383

 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, CBE are drawn terminated by the other circumference: shew that the arc DE is invariable.

222. Through a point C 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