Two Non-Intersecting Circles have Four Common Tangents

Theorem

Let $C_1$ and $C_2$ be circles embedded in the plane such that:

$C_1$ and $C_2$ do not intersect

one is not inside the other.

Then there are $4$ common tangents to $C_1$ and $C_2$:

$2$ of the common tangents have the circles on the same side of the tangent: the external tangents

$2$ of the common tangents have the circles on opposite sides of the tangent: the internal tangents.

Proof

This theorem requires a proof. In particular: It's visually obvious, but will need to be proved algebraically. That will probably result in a quartic. 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.

Also see

• Definition:External Tangent
• Definition:Internal Tangent

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): common tangent
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): common tangent