Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/381

 given circle, the sum of the two is least which make equal angles with the tangent at the point of concourse.

201. C is the centre of a given circle, CA a radius, B a point on a radius at right angles to CA; join AB and produce it to meet the circle again at D, and let the tangent at D meet CB produced at E: shew that BDE is an isosceles triangle.,

202. Let the diameter BA of a circle be produced to P, so that AP equals the radius; through A draw the tangent AED, and from P draw PEC touching the circle at C and meeting the former tangent at E; join BC and produce it to meet AED at D: then will the triangle DEC be equilateral.

III. 20 to 22.

203. Two tangents AB, AC are drawn to a circle; D is any point on the circumference outside of the triangle ABC: shew that the sum of the angles ABD and ACD is constant.

204. P, Q are any points in the circumferences of two segments described on the same straight line AB, and on the same side of it; the angles PAQ, PBQ are bisected by the straight lines AR, BR meeting at R: shew that the angle ARB is constant.

205. Two segments of a circle are on the same base AB, and P is any point in the circumference of one of the segments; the straight lines APD, BPC are drawn meeting the circumference of the other segment at D and C; AC and BD are drawn intersecting at Q. Shew that the angle AQB is constant.

206. APB is a fixed chord passing through P a point of intersection of two circles AQP, PBR; and QPR is any other chord of the circles passing through P: shew that AQ and RB when produced meet at a constant angle.

207. AOB is a triangle; C and D are points in BO and AO respectively, such that the angle ODC is equal to the angle OBA: shew that a circle may be described round the quadrilateral ABCD.