A) Let z1 and z2 be two points on a circle C. Let z3 and z4 be symmetric with respect to the circle. Show that the cross ratio (z1,z2,z3,z4) has absolute value 1.
b)Let ad-bc=1, c not zero and consider T(z)=(az+b)/(cz+d). Show that it increases lengths and areas inside the circle|cz+d|=1 and decreases lengths and areas outside the circle|cz+d|=1
Circles and Cross Ratios are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.