Equivalence of Definitions of Convex Polygon

Theorem

The following definitions of the concept of Convex Polygon are equivalent:

Definition 1

Let $P$ be a polygon.

$P$ is a convex polygon if and only if:

For all points $A$ and $B$ located inside $P$, the line $AB$ is also inside $P$.

Definition 2

Let $P$ be a polygon.

$P$ is a convex polygon if and only if:

every internal angle of $P$ is not greater than $180 \degrees$.

Definition 3

Let $P$ be a polygon.

$P$ is a convex polygon if and only if:

the region enclosed by $P$ lies entirely on the same side of each side of $P$

Proof

$(1)$ implies $(2)$

Let $P$ be a convex polygon by definition $1$.

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Thus $P$ is a convex polygon by definition $2$.

$\Box$

$(2)$ implies $(1)$

Let $P$ be a convex polygon by definition $2$.

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Thus $P$ is a convex polygon by definition $1$.

$\Box$

This needs considerable tedious hard slog to complete it. In particular: definition 3 etc. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.