(i) Letz=eiθ−eiφeiθ+eiφ,where θ and φ are real, and θ−φ=2nπ for any integer n. Show thatz=icot21(φ−θ)and give expressions for the modulus and argument of z.
(ii) The distinct points A and B lie on a circle with radius 1 and centre O. In the complex plane, A and B are represented by the complex numbers a and b, and O is at the origin. The point X is represented by the complex number x, where x=a+b and a+b=0. Show that OX is perpendicular to AB.
If the distinct points A, B and C in the complex plane, which are represented by the complex numbers a, b and c, lie on a circle with radius 1 and centre O, and h=a+b+c represents the point H, then H is said to be the orthocentre of the triangle ABC.
(iii) The distinct points A, B and C lie on a circle with radius 1 and centre O. In the complex plane, A, B and C are represented by the complex numbers a, b and c, and O is at the origin. Show that, if the point H, represented by the complex number h, is the orthocentre of the triangle ABC, then either h=a or AH is perpendicular to BC.
(iv) The distinct points A, B, C and D (in that order, anticlockwise) all lie on a circle with radius 1 and centre O. The points P, Q, R and S are the orthocentres of the triangles ABC, BCD, CDA and DAB, respectively. By considering the midpoint of AQ, show that there is a single transformation which maps the quadrilateral ABCD on to the quadrilateral QRSP and describe this transformation fully.
