# Fixed Points of Projection in Plane

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## Theorem

Let $M$ and $N$ be distinct lines in the plane.

Let $\pr_{M, N}$ be the **projection on $M$ along $N$**:

- $\forall x \in \R^2: \map {\pr_{M, N} } x =$ the intersection of $M$ with the line through $x$ parallel to $N$.

Then $M$ is the set of fixed points of $\pr_{M, N}$ in the sense that:

- $x \in M$

- $\map {\pr_{M, N} } x = x$

## Proof

### Sufficient Condition

Let $x \in M$.

Let $\LL$ be the straight line through $x$ which is parallel to $N$.

As $x \in M$ it follows that $x$ is on the intersection of $M$ with $\LL$.

Hence by definition:

- $\map {\pr_{M, N} } x = x$

$\Box$

### Necessary Condition

Again, let $\LL$ be the straight line through $x$ which is parallel to $N$.

Let $\map {\pr_{M, N} } x = x$.

Then by definition $x$ is on the intersection of $M$ with $\LL$.

Hence by definition of intersection:

- $x \in M$.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.5$