# Notes - Complex Analysis MT23, Geometry

### Flashcards

Suppose $a, b \in \mathbb C$ are distinct. What does the locus of

\[|z - a| = |z - b|\]
represent?

A line.

What are the two ways of representing a circle in $\mathbb C$ as a locus?

\[|z - a| = r\]
\[|z - a| = \lambda |z - b|\]

Circles can be described in $\mathbb C$ as $ \vert z - a \vert = \lambda \vert z - b \vert $. How does this translate to a fact about standard Euclidean geometry?

The set of all points $P$ such that the ratio of the distances $PA / PB = \lambda$ is a circle.

### Proofs

Prove that every line in $\mathbb C$ can be written as the locus

\[|z - a| = |z - b|\]
for some $a, b \in \mathbb C$.

Todo.

Prove that every circle in $\mathbb C$ can be written as

\[|z - a| = \lambda |z - b|\]
for some $a, b \in \mathbb C$ and some $\lambda \in (0, \infty)$.

Todo.