Geometry: Equilateral Tri. and Circumcirles

  • Thread starter wubie
  • Start date
  • Tags
    Geometry
In summary, the conversation discusses the confusion over the term "smaller arc" in reference to a point on the circumcircle of an equilateral triangle. The experts clarify that when two points on a circle are mentioned, one arc is always smaller than the other and in this case, the smaller arc is the one being referred to as BC. This confusion was due to trying to be too precise in the wording.
  • #1
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.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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?
 
  • #4
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".
 
  • #5
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.)
 
  • #6
Great. Thanks Doc Al, HallsofIvy.

Cheers.
 

1. What is an equilateral triangle?

An equilateral triangle is a triangle with all three sides of equal length. This means that all angles in an equilateral triangle are also equal, measuring 60 degrees each.

2. How do you find the area of an equilateral triangle?

The formula for finding the area of an equilateral triangle is A = (sqrt(3)/4) * s^2, where s is the length of one side. This means you square the length of one side and multiply it by the square root of 3 divided by 4.

3. What is a circumcircle?

A circumcircle is a circle that passes through all three vertices of a triangle. In an equilateral triangle, the circumcircle will have the same center as the triangle and its radius will be equal to the length of any side.

4. How do you find the radius of the circumcircle of an equilateral triangle?

The radius of the circumcircle of an equilateral triangle can be found using the formula r = s/(sqrt(3)), where s is the length of one side. This means you divide the length of one side by the square root of 3.

5. Can an equilateral triangle have a circumcircle with a radius of 0?

Yes, an equilateral triangle can have a circumcircle with a radius of 0. This occurs when the triangle's vertices are all located at the center of the circle, making the radius equal to the distance between any two vertices, which is 0.

Similar threads

  • Introductory Physics Homework Help
Replies
5
Views
2K
  • Introductory Physics Homework Help
Replies
13
Views
1K
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
934
  • General Math
Replies
4
Views
749
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
12
Views
2K
  • Introductory Physics Homework Help
Replies
6
Views
967
Replies
8
Views
928
  • Introductory Physics Homework Help
Replies
2
Views
3K
Back
Top