# ABC is an equilateral tiangle.P is any point on the arc BC.Prove AP=BP+PC http://www.flickr.com/photos/78780315@N06/7963176188/in/photostream

lfryerda | Certified Educator

Since the points ABPC lie on a common circle, then the quadrilateral is cyclic (see link below).  Cyclic quadrilaterals obey Ptolemy's theorem, so we have :

`AP cdot BC= AC cdot BP+PC cdot AB`

but since triangle ABC is equilateral, each side is the same.  Let

`x=AB=AC=BC`

then Ptolemy's Theorem becomes

`AP cdot x=x cdot BP+PC cdot x`   divide both sides by x to get

`AP=BP+PC`

The theorem has been proven.

topperoo | Student
the link is this http://www.flickr.com/photos/78780315@N06/7963176188/in/ photostream
The text passage contains a geometric proof for a statement about an equilateral triangle and a point on its circumcircle. The proof uses Ptolemy's theorem to show that for an equilateral triangle ABC and a point P on the arc BC, the length of AP is equal to the sum of the lengths of BP and PC.
no_question

the passage does not contain a mathematical question but rather a proof of a geometric statement. therefore, there is no question and no answer provided in the text.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# ABC is an equilateral tiangle.P is any point on the arc BC.Prove AP=BP+PC http://www.flickr.com/photos/78780315@N06/7963176188/in/photostream

lfryerda | Certified Educator

Since the points ABPC lie on a common circle, then the quadrilateral is cyclic (see link below).  Cyclic quadrilaterals obey Ptolemy's theorem, so we have :

`AP cdot BC= AC cdot BP+PC cdot AB`

but since triangle ABC is equilateral, each side is the same.  Let

`x=AB=AC=BC`

then Ptolemy's Theorem becomes

`AP cdot x=x cdot BP+PC cdot x`   divide both sides by x to get

`AP=BP+PC`

The theorem has been proven.

topperoo | Student
the link is this http://www.flickr.com/photos/78780315@N06/7963176188/in/ photostream
user
The text passage contains a geometric proof for a statement about an equilateral triangle and a point on its circumcircle. The proof uses Ptolemy's theorem to show that for an equilateral triangle ABC and a point P on the arc BC, the length of AP is equal to the sum of the lengths of BP and PC.

TYPE: NO_QUESTION

The passage does not contain a mathematical question but rather a proof of a geometric statement. Therefore, there is no question and no answer provided in the text.