Compute the center of Dih(2n)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.7
Let $n \in \mathbb{Z}$ with $n \geq 3$. Prove the following.
(1) $Z(D_{2n}) = 1$ if $n$ is odd.
(2) $Z(D_{2n}) = \{ 1, r^k \}$ if $n = 2k$ is even.

Solution:
(1) This is the content of Exercise 1.2.4.
(2) This is the content of Exercise 1.2.5.