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