Using Ptolemy's Theorem to prove simple (yet unique) cases?

[theJorge551]

Problem: Point P is on arc AB of the circumcircle of regular hexagon ABCDEF. Prove that PD + PE = PA + PB + PC + PF.

I'm aware that I'm supposed to use Ptolemy's theorem, which states that

if a quadrilateral ABCD is cyclic, then AC x BD = AB x CD + AD x BC.

I've drawn the hexagon and it seems like an unreasonably long proof is required if I wanted to prove it based on various quadrilaterals inherent within the circle, but I've got a hunch that I can use the analog for a triangle:

If point P is on arc AB of the circumcircle of equilateral triangle ABC, then PC = PA + PB,

which I've already proved in a separate question.

Any tips for getting the ball rolling for this proof?

Also, there's another question that I have based on using Ptolemy's theorem:

Let P be a point in the plane of triangle ABC such that the segments PA, PB, and PC are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to PA. Prove that angle BAC is acute.

For this problem, primarily, I'm having a difficult time determining how to draw the figure. I'm sure that once I am able to, I'll be able to work through to a solution using Ptolemy's inequality, but how should I go about making a figure for this? I find the wording to be a bit abstract compared to how it usually is in these types of problems.

 

[NascentOxygen]

I'm aware that I'm supposed to use Ptolemy's theorem, which states that

AC ?? BD = AB ?? CD + AD ?? BC

Whatever operators you have there, they don't show up for me. (Other than as 0x95).

 

[theJorge551]

Edited out the weird symbols and replaced them. Should be all clear now. Sorry for the confusion.

 

[NascentOxygen]

Also, there's another question that I have based on using Ptolemy's theorem:For this problem, primarily, I'm having a difficult time determining how to draw the figure.

There must be an error somewhere. How can 3 lines radiating from a point form the sides of a triangle? And what is meant by "the side congruent to PA"? Are there meant to be two triangles?

Too many errors.

 

[theJorge551]

There must be an error somewhere. How can 3 lines radiating from a point form the sides of a triangle? And what is meant by "the side congruent to PA"? Are there meant to be two triangles?

Too many errors.

I went over the problem today, and it turns out that the wording was fallacious. I've seen the solutions for both of these problems now (the way they were meant to be interpreted) and I no longer need help with them...but the ones in my other thread are much more pressing. :P