Dem.—Let O, O′ be the centres of the two fixed circles; O′′ the centre of the variable circle; A, B the points of contact. Let AB and OO′ meet in C, and cut the fixed circles again in the points A′, B′ respectively. Join A′O, AO, BO′. Then AO, BO′ meet in O′′ [III. xi.]. Now, because the triangles OAA′, O′′AB are isosceles, the angle O′′BA = O′′AB = OA′A. Hence OA′ is parallel to O′B; therefore OC : O′C :: OA′ : O′B; that is, in a given ratio. Hence C is a given point. 28. If DD′ be the common tangent to the two circles, DD′2 = AB′.A′B. 29. If R denote the radius of O′′ and ρ, ρ′, the radii of O, O′, DD′2 : AB2 :: (R±ρ)(R±ρ′) : R2, the choice of sign depending on the nature of the contacts. This follows from 28. 30. If four circles be tangential to a fifth, and if we denote by 12 the common tangent to the first and second, &c., then 31. The inscribed and escribed circles of any triangle are all touched by its nine-points circle. 32. The four triangles which are determined by four points, taken three by three, are such that their nine-points circles have one common point. 33. If a, b, c, d denote the four sides, and D, D′ the diagonals of a quadrilateral; prove that the sides of the triangle, formed by joining the feet of the perpendiculars from any of its angular points on the sides of the triangle formed by the three remaining points, are proportional to the three rectangles ac, bd, DD′. 34. Prove the converse of Ptolemy’s theorem (see xvii., Ex. 13). 35. Describe a circle which shall—(1) pass through a given point, and touch two given circles; (2) touch three given circles. 36. If a variable circle touch two fixed circles, the tangent to it from their centre of similitude, through which the chord of contact passes (27), is of constant length. 37. If the lines AD, BD′ (see fig., Ex. 27) be produced, they meet in a point on the circumference of O′′, and the line O′′P is perpendicular to DD′. 38. If A, B be two fixed points on two lines given in position, and A′, B′ two variable points, such that the ratio AA′ : BB′ is constant, the locus of the point dividing A′B′ in a given ratio is a right line. 39. If a line EF divide proportionally two opposite sides of a quadrilateral, and a line GH the other sides, each of these is divided by the other in the same ratio as the sides which determine them. 40. In a given circle inscribe a triangle, such that the triangle whose angular points are the feet of the perpendiculars from the extremities of the base on the bisector of the vertical angle, and the foot of the perpendicular from the vertical angle on the base, may be a maximum. 41. In a circle, the point of intersection of the diagonals of any inscribed quadrilateral coincides with the point of intersection of the diagonals of the circumscribed quadrilateral, whose sides touch the circle at the angular points of the inscribed quadrilateral.