Difficult circle geometry question, proof sum of 2 sides ratio equals 1

Question

I have been working on this question for hours on end but not even come close to solving. I have found 2 pairs of similar triangles and an isosceles triangle, and tried to equate the ratio of sides, but cannot prove the question provided. Any help is greatly appreciated. Thanks in advance.

Answer 1

It is fairly straightforward to prove that $PAC$ and $PBA$ are similar. Note that the angle bisector is the bisector in both triangles, so we can see that the diagram $PAEC$ is similar to $PBDA$. So we obtain $\frac{EC}{AC} = \frac{DA}{AB}$, from which the result follows.