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

Expert Answers
lfryerda eNotes educator| 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