30. Place a given triangle so that its three sides shall pass through three given points. 31. Place a given triangle so that its three vertices shall lie on three given lines. 32. Construct the greatest triangle equiangular to a given one whose sides shall pass through three given points. 33. Construct the least triangle equiangular to a given one whose vertices shall lie on three given lines. 34. Construct the greatest triangle equiangular to a given one whose sides shall touch three given circles. 35. If two sides of a given triangle pass through fixed points, the third touches a fixed circle. 36. If two sides of a given triangle touch fixed circles, the third touches a fixed circle. 37. Construct an equilateral triangle having its vertex at a given point, and the extremities of its base on a given circle. 38. Construct an equilateral triangle having its vertex at a given point, and the extremities of its base on two given circles. 39. Place a given triangle so that its three sides shall touch three given circles. 40. Circumscribe a square about a given quadrilateral. 41. Inscribe a square in a given quadrilateral. 42. Describe circles—(1) orthogonal (cutting at right angles) to a given circle and passing through two given points; (2) orthogonal to two others, and passing through a given point; (3) orthogonal to three others. 43. If from the extremities of a diameter AB of a semicircle two chords AD, BE be drawn, meeting in C, AC.AD + BC.BE = AB2. 44. If ABCD be a cyclic quadrilateral, and if we describe any circle passing through the points A and B, another through B and C, a third through C and D, and a fourth through D and A; these circles intersect successively in four other points E, F, G, H, forming another cyclic quadrilateral. 45. If ABC be an equilateral triangle, what is the locus of the point M, if MA = MB + MC? 46. In a triangle, given the sum or the difference of two sides and the angle formed by these sides both in magnitude and position, the locus of the centre of the circumscribed circle is a right line. 47. Describe a circle—(1) through two given points which shall bisect the circumference of a given circle; (2) through one given point which shall bisect the circumference of two given circles. 48. Find the locus of the centre of a circle which bisects the circumferences of two given circles. 49. Describe a circle which shall bisect the circumferences of three given circles. 50. AB is a diameter of a circle; AC, AD are two chords meeting the tangent at B in the points E, F respectively: prove that the points C, D, E, F are concyclic. 51. CD is a perpendicular from any point C in a semicircle on the diameter AB; EFG is a circle touching DB in E, CD in F, and the semicircle in G; prove—(1) that the points A, F, G are collinear; (2) that AC = AE.