text: # 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
explanation: The document contains a math question and its answer. The question is about proving a property of an equilateral triangle inscribed in a circle, and the answer provides a proof using Ptolemy's theorem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its answer. The question is about proving a property of an equilateral triangle inscribed in a circle, and the answer provides a proof using Ptolemy's theorem.
  
  Label: question-and-answer