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

• theJorge551
In summary, the problem with these questions is that they are written in a confusing and abstract manner, and it is difficult to follow the instructions.
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.

Last edited:
theJorge551 said:
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).

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

theJorge551 said:
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.

NascentOxygen said:
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

## 1. What is Ptolemy's Theorem?

Ptolemy's Theorem is a mathematical theorem that relates the sides and diagonals of a cyclic quadrilateral. It states that in a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides.

## 2. How is Ptolemy's Theorem used to prove simple cases?

Ptolemy's Theorem can be used to prove simple cases by applying it to cyclic quadrilaterals with specific properties, such as when all the sides are equal or when the diagonals are perpendicular. By using these specific cases and applying Ptolemy's Theorem, we can prove the desired result.

## 3. What are some real-life applications of Ptolemy's Theorem?

Ptolemy's Theorem has many practical applications, such as in navigation and surveying. It can be used to calculate distances and angles in maps and to determine the position of objects in space.

## 4. Can Ptolemy's Theorem be applied to non-cyclic quadrilaterals?

No, Ptolemy's Theorem can only be applied to cyclic quadrilaterals, which are quadrilaterals with all four vertices lying on a circle.

## 5. How does Ptolemy's Theorem relate to other theorems in geometry?

Ptolemy's Theorem is related to other theorems in geometry, such as the Pythagorean Theorem and the Law of Cosines. It can also be used in conjunction with other theorems, such as the Intersecting Chords Theorem and the Inscribed Angle Theorem, to prove more complex cases.

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