Prove that PT is a tangent to the circle ADT at T. DA is a tangent to circle ACBT at A. Show that ATP = ADT.

To prove that PT is a tangent to the circle ADT at T, we need to show that the angle between the line PT and the chord AT (of the circle ADT) is equal to the angle subtended by the chord AT in the alternate segment, which is $\angle ADT$. That is, we need to prove $\angle ATP = \angle ADT$.

Step 1: Use the Tangent-Chord Theorem for circle ACBT.
Given that $DA$ is a tangent to the circle $ACBT$ at $A$, and $\angle A_1 = x$.
According to the Tangent-Chord Theorem, the angle between the tangent ($DA$) and a chord ($AT$) is equal to the angle in the alternate segment.

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