The Eyeball Theorem with 8 Proofs

Theorem 

Let there be two circles $C(A,R_A)$ and $C(B,R_B)$, one with center $A$ and radius $RA$, the other with center $B$ and radius $R_B$. Assume the tangents from $A$ to $C(B,R_B)$ intersect $C(A,R_A)$ at points $P$ and $Q$, whereas the tangents from $B$ to $C(A,R_A)$ intersect $C(B,R_B)$ at points $R$ and $S$.

Proof 1

Proof 2

Proof 2'

Proof 3

Proof 3'

Proof 4

Proof 5

Proof 6