In figure two circles intersect at two points A and B. From a point P on a circle, two line segments PAC and PBD are drawn intersecting the other circle at the points C and D. Prove that CD is parallel to the tangent at P.
Given: Two circles intersect at A and B.
To prove: CD || tangent at P.
Proof: Join AB. Let XY be the tangent at P. Then by alternate segment theorem,
∠ APX = ∠ ABP ……………(i)
Next, ABCD is a cyclic quadrilateral, therefore, by the theorem sum of the opposite angles of a quadrilateral is 180°
∠ ABD + ∠ ACD = 180°
Also ∠ ABD = ∠ ABP = 180° (Linear Pair)
∴ ∠ ACD = ∠ ABP ...........(ii)
From (i) and (ii),
∠ ACD = ∠ APX
∴ XY || CD (Since alternate angles are equal).
To prove: CD || tangent at P.
Proof: Join AB. Let XY be the tangent at P. Then by alternate segment theorem,
∠ APX = ∠ ABP ……………(i)
Next, ABCD is a cyclic quadrilateral, therefore, by the theorem sum of the opposite angles of a quadrilateral is 180°
∠ ABD + ∠ ACD = 180°
Also ∠ ABD = ∠ ABP = 180° (Linear Pair)
∴ ∠ ACD = ∠ ABP ...........(ii)
From (i) and (ii),
∠ ACD = ∠ APX
∴ XY || CD (Since alternate angles are equal).
12