In this figure a, b, c, d, e, f are two sets of collinear points. Prove that ad ll cf.

The figure is given below:

In this figure, ABED is a cyclic quadrilateral.

∠DAB = ∠ BED ( as opposite angles of a cyclic quadrilateral are supplementary) ...(1)

BCFE is also a cyclic Quad.

∴ ∠BCF = ∠ BEF(as opposite angles of a cyclic quadrilateral are supplementary)…(2)

when we add these angles

∠DAB + ∠ BED + ∠BCF + ∠ BEF = 360°…(1)

∠BED + ∠ BEF = 180°

⇒ ∠ DAB + ∠ BCF = 180°

But they are co-interior angles and if co-interior angles sum up to 180° they are parallel.

So AD parallel to CF.