11.10: Finding the point E is the right way to go.  Instead of proving equality, you use Triangle Inequality.

11.11: Proving Ptolemy's Theorem and Simson's Theorem are equivalent.  You need to formulate the problem first.  Say you are trying to prove Simson's Theorem.  Look at the conditions of Simson's Theorem.  Use those, and apply Ptolemy's Theorem in the process (we are assuming Ptolemy's Theorem is true, and prove Simson's Theorem).  Then you need to prove the other direction: assuming Simson's Theorem is right, prove Ptolemy's.