In figure, two circles with centres O, O' touch externally at a point A. A line through A is drawn to intersect these circles at B and C. Prove that the tangents at B and C are parallel.  [Hint: Prove that ∠ OBA = ∠ O' CA]
        

Given: Two circles with centers O, O' touch externally at a point A. A line through A is 
drawn to intersect these circles in B and C.
To prove: Tangents B and C are parallel.
Proof:
 
In Δ AOB,
∠ OBA = ∠ OAB       (OB = OA, opposite angles of equal sides) ………………..(i)
In Δ AO'C
∠ O'AC = ∠ O'CA    (O'A = O'C, opposite angles of equal sides) ………………..(ii)
∠ OAB = ∠ O'AC (Vertically opposite angles) ………………..(iii)
From (i), (ii) and (iii)
∠ OBA = ∠ O'CA ……………………(iv)
Since the line joining the centre and the tangent at the point of contact is 90°.
∠ OBX = ∠ O'CY = 90° ………………….(v)
Adding (iv) and (v)
∠ OBX + ∠ OBA = ∠ O'CY + ∠ O'CA
∠ XBA = ∠ YCA
Since alternate angles are equal, tangents at B and C are parallel.