The diagram shows a point $A$ on the circle and $XAY$ is a tangent to the circle. Points $S$, $B$ and $C$ lie on the circle. The chords $AB$ and $SC$ intersect at $T$ and angle $ACB$ = angle $ATC$.

1. Identify the common angle between the two triangles

In $\triangle ABC$ and $\triangle ACT$,
$\angle BAC = \angle TAC$ (Common angle)

2. Use the given information to establish a second pair of equal angles and conclude similarity

$\angle ACB = \angle ATC$ (Given)
Since two pairs of corresponding angles are equal, $\triangle ABC$ and $\triangle ACT$ are similar (AA Similarity).

1. Use the similarity of triangles from part (a) to set up a ratio of corresponding sides.

Since $\triangle ABC$ is similar to $\triangle ACT$ (from part (a)), the ratio of their corresponding sides is equal:
$\frac{AC}{AT} = \frac{AB}{AC}$

2. Express $AC^2$ in terms of $AT$ and $AB$.

Cross-multiplying the ratio gives:
$AC^2 = AT \times AB$

3. Substitute $AB = AT + TB$ to obtain the required expression.

Since $T$ lies on the chord $AB$, we have $AB = AT + TB$. Substituting this into the equation:
$AC^2 = AT \times (AT + TB)$
$AC^2 = AT^2 + AT \times TB$
$AC^2 - AT^2 = AT \times TB$ (Shown)

1. Identify equal angles using the alternate segment theorem

By the alternate segment theorem, the angle between the tangent $XAY$ and the chord $AC$ is equal to the angle in the alternate segment.
$\angle YAC = \angle ABC$

2. Use the similarity from part (a) to relate angles

From part (a), $\triangle ABC$ is similar to $\triangle ACT$.
Therefore, corresponding angles are equal: $\angle ABC = \angle ACT$.

3. Conclude that the lines are parallel using alternate angles

Since $\angle YAC = \angle ABC$ and $\angle ABC = \angle ACT$, we have $\angle YAC = \angle ACT$.
Notice that $\angle ACT$ is the same as $\angle ACS$ (since $T$ lies on $SC$).
Thus, $\angle YAC = \angle ACS$.
These are alternate interior angles formed by the transversal $AC$ intersecting lines $XY$ and $SC$.
Since the alternate angles are equal, the lines $SC$ and $XY$ are parallel.