# Fractional Transformations, Cross Ratios and Conformal Mapping

1. a) Let z1,z2,z3,z4 lie on a circle. Show that z1,z3,z4 and z2,z3,z4 determine the same orientation iff (z1,z2,z,3,z4)>0

b) Let z1,z2,z3,z4 lie on a circle and be consecutive vertices of a quadrilateral. Prove that |z1-z3|*|z2-z4|=|z1-z2|*|z3-z4|+|z2-z3|*|z1-z4|

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#### Solution Preview

a) Let

a = Arg (z1,z3,z4) = Arg [(z4-z1)/(z3-z1)]

= Arg (z4-z1) - Arg (z3-z1)

= anticlockwise turn from >-Z1Z3-> to >-Z1Z4->

(negative means clockwise turn)

b = Arg (z2,z3,z4)

= anticlockwise turn from >-Z2Z3-> to >-Z2Z4->

From school geometry (and taking care of the clockwise /

anticlockwise senses) we get two cases:

Case 1

b = a (when Z1, Z2 are on same side of chord Z3Z4

i.e. when Z1,Z3,Z4 and Z2,Z3,Z4 have same orientation)

Case 2

b = -(pi - a) (when ...

#### Solution Summary

Fractional Transformations, Cross Ratios and Conformal Mapping are investigated. The solution is detailed and well presented.