Q:

Complex conjugate graphing translation

A)

So far, I know that these triangulars are equilateral and each side length is sqrt(10) How would I find C1 * C2? Should I translate the coordinates?
B)
For this problem f(z) and z are perpendicular. I'm trying to find f(z) in terms of az+b such that a and b are complex numbers. I got zi-2i+1 for the value of f(z) from f(z)-(2+i). How do I continue?

A:

Hint:   a rotation by angle $\,\alpha\,$ counterclockwise in the complex plane is equivalent with multiplication by $\,e^{i \alpha}=\cos \alpha + i \sin \alpha\,$. In the first problem, this translates to:
$$
c_1 - (3+2i) = e^{i \pi/3} \big((6+i)-(3+2i)\big) = \frac{1+i\sqrt{3}}{2} \cdot (3-i)
$$

[ EDIT ]   Since what's asked is just the product $\,c_1 \cdot c_2\,$, the calculations can be simplified somewhat to get $\,c_1 \cdot c_2\,$ directly, without determining $\,c_1,c_2\,$ individually.
Let $\,a=3+2i\,$, $\,u=(6+i)-(3+2i)=3-i\,$ and $\,\omega = e^{i \pi/3} = \frac{1+i\sqrt{3}}{2}\,$. Then $\,\bar \omega = e^{-i \pi/3}\,$, and the equations can be written as:
$$
c_1 - a = \omega u \\
c_2 - a = \bar \omega u
$$
It follows that:
$$
c_1 \cdot c_2=(a+\omega u)(a + \bar \omega u) = a^2 + |\omega|^2 u^2 + au(\omega+\bar\omega) = a^2 + u^2 + 2 au \operatorname{Re}(\omega)
$$
Since $\,2 \operatorname{Re}(\omega) = 1\,$, the calculations no longer involve any $\,\sqrt{3}\,$ terms.

