# Geometry: Equilateral Tri. and Circumcirles

wubie
Hello,

First I will post my question.

ABC is an equilateral triangle, and P is a point on the smaller arc BC of the cirumcircle of the triangle ABC. Prove that PA = PB + PC.

What is confusing me is the part

smaller arc BC of the cirumcircle of the triangle ABC

If ABC is an equilateral triangle, why would the arc BC be smaller than the arcs AB or AC? I would think that all the arcs would be equal. What the hey? What am I missing here?

Any help is appreciated.

Thankyou.

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Doc Al
Mentor
Originally posted by wubie
If ABC is an equilateral triangle, why would the arc BC be smaller than the arcs AB or AC? I would think that all the arcs would be equal. What the hey? What am I missing here?
Two points (B & C) on a circle delimit two arcs, not three (you are splitting the longer arc). In this case, the short one is B-P-C; the long one goes B-A-C.

wubie
Hello Doc Al,

I am not sure what you are saying.

If I was to construct an equilateral triangle, then construct each side's perpendicular bisector, then construct the circumcircle, I would had an equalateral triangle inside the circumcircle.

In this case, all vertices of the triangle ABC would be points on the circumcircle. Arc AB = BC = CA since the triangle is equilateral.

Now there would be two paths of "travel" from B to C, one of which would pass through point A. That is, one path would go directly to C (the smaller arc) and the other path would go from B to A then to C (the larger arc).

Is this what you mean by BC delimiting two arcs; one small and one large?

HallsofIvy
Homework Helper
Forget the triangle. Two points on a circle divide it into two arcs. Unless the two points are 180 degrees apart, one of the arcs is smaller than the other. THAT'S the one they are talking about.

I will confess I was taken aback by that myself. Normally when you say "the arc BC", the smaller of the two arcs is what is meant. I suspect this was a case of confusing by trying to be too precise.

The "smaller arc" in this case is precisely what you think of as "the arc BC".

Doc Al
Mentor
Originally posted by wubie
Is this what you mean by BC delimiting two arcs; one small and one large?
You got it. Sorry if I wasn't clear enough. (The wording took me a minute to figure out at first too.)

wubie
Great. Thanks Doc Al, HallsofIvy.

Cheers.