**The Common Tangents Theorem**

Suppose that **C** and **D** are distinct conics.

Then the common tangents (if any) to **D** and **C**

occur at the points
where **C** intersects the dual of **D** with respect to **C**.

**Proof**

Throughout the proof **duality** means **duality with respect to C.**

Let **D'** be the dual of **D**. Then, of course, **D** is the dual of **D'**.

Suppose first that **D'** meets **C** at T.

As T is on **C**, the dual of T is the tangent to **C** at T.

As **D** is the dual of **D'**, the dual of T is a tangent to **D**,

so is a common tangent to **C** and **D**.

Suppose now that there is a common tangent **L** to **C** and
**D**, meeting **C** at T, say.

As **L** is the tangent to **C **at T, T is the dual of **L**.

As **L** is also a tangent to **D**, T, the dual of **L**, must lie on **D'**, the dual of **D**.

Thus T lies on the intersection of **C** and the dual of **D**.