I understand that Möbius transformations are angle preserving transformations.
Knowing this, my professor asked us to think about how the image of equilateral triangle is not an equilateral triangle in general. Explain why. He asked us to give an example with the vertices of an equilateral triangle, A, B, C, and a Möbius transformation T so that the images T(A), T(B), and T(C), do not form an equilateral triangle.
Can anyone help me with this? I am trying to study for my geometry final. He will release an answer key for us on December 12th, but I'm just going through practice questions beforehand.
Thanks.
Möbius transformations transform straight lines to straight lines or circles. So, the image of your triangle is a triangle with circular arc sides, which DO meet at the same angles that the sides of the triangle did. As for your second question, take your favorite Möbius transformation and your favorite equilateral triangle and compute.