Let AB be the tangent to the circle at point P with center O.
To prove: PQ passes through the point O.
Construction: Join OP.
Through O, draw a straight line CD parallel to the tangent AB.
Proof: Suppose that PQ doesn’t passes through point O.
PQ intersect CD at R and also intersect AB at P
AS, CD || AB. PQ is the line of intersection.
∠ORP = ∠RPA (Alternate interior angles)
but also.
∠RPA = 90° (OP ⊥ AB)
\(\Rightarrow\) ∠ORP = 90°
∠ROP + ∠OPA = 180° (Co interior angles)
\(\Rightarrow\) ∠ROP + 90° = 180°
\(\Rightarrow\) ∠ROP = 90°
Thus, the △ORP has 2 right angles i.e., ∠ORP and ∠ROP which is not possible Hence, our supposition is wrong
∴ PQ passes through the point O.
