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.