2025 CGSS PRELIMS P1 Q14

In the figure below, points \(A\), \(B\) and \(C\) lie on the circle. \(AC\) is a diameter of a circle with centre \(O\). The tangent to the circle at \(X\) meets \(AC\) produced at \(Y\). Chord \(XB\) intersects \(AC\) at \(F\). \(XC\) bisects the angle of \(YXB\).

1. Prove that \(CX = CB\).[2]
2. Prove that triangle \(AXC\) is similar to triangle \(BFC\).[3]
3. Hence, prove that \(AC \times FC = BC^2\).[2]