2026 MSHS P2 Q7

In the diagram, \(A\), \(B\), \(C\) and \(D\) lie on a circle. \(XT\) is the tangent to the circle at \(A\) and the line \(DB\) extended meets the tangent at \(T\). Points \(E\) and \(F\) lie on \(XAT\) and \(DBT\) respectively. \(EF\) is parallel to \(AB\) and point \(E\) is the mid-point of \(AT\). It is given that \(AB\) bisects angle \(CAT\) and \(AD\) bisects angle \(CAX\).

1. Prove that \(AB=BC\).[2]
2. Prove that \(BD\) is the diameter of the circle.[3]
3. 1. Prove that triangle \(TAD\) is similar to triangle \(TBA\).[3]
   2. Hence, show that \(BT\times DT=4ET^2\).[2]