How Do Three Tangents Form an Equilateral Triangle Around a Circle?

[Saracen Rue]

Homework Statement

3 tangents intersect a circle of unknown proportions, forming an equilateral triangle.

a) Represent this on a Cartesian Plane, with the center of the circle situated at the origin
b) Prove that the intersections with the circle are the midpoints of the tangents (assume the tangents terminate once they form the triangle).
c) Calculate the percentage area of the triangle which is not part of the circle

Homework Equations

No equations were given. I was, however, told I needed to know basic area formulae and have an understanding of the unit circle.

The Attempt at a Solution

I can do part a) easily enough, but I can't figure out how to do either of the next two questions. My guess is that I need to find a way to represent the length of the tangent's point of interception with the circle to the end of the tangent, then prove that the full length of the tangent is double that. How I would go about doing that - assuming it is even the correct way to be think about this - is beyond me. Similarly with the last part, I need to find a way to represent the area of the circle in terms of the area of the triangle, but how I would do this with no numbers given at all is unbeknownst to me.

 

[HallsofIvy]

What do you mean by "three tangents that intersect the circle"? The lines that are tangent to that circle? So "intersect" just refers to the point of tangency?

Setting up a coordinate system with the center of the circle at the origin, the circle has equation x^2+ y^2= R^2 for some constant R. We can further choose the coordinate system so that the intersection of one pair of the tangent lines is on the positive y-axis. In this case, one of the tangent lines is given by y= -R.

 

[SteamKing]

You are given an important clue: the three tangents are situated such that an equilateral triangle is formed. What are the angles of an equilateral triangle? Also, use the fact that a radius drawn from the center of the circle to the point of tangency is perpendicular to the tangent at that point.

 

[Saracen Rue]

Okay, I'm still a bit confused on how I would go about doing the second part, but I did manage to get an answer of 39.54% for the third part. Can anyone validate if this is correct or not?

 

[Bill_Nye_Fan]

Okay, I'm still a bit confused on how I would go about doing the second part, but I did manage to get an answer of 39.54% for the third part. Can anyone validate if this is correct or not?

I, too, got 39.54% for the remaining area of the triangle which is not part of the circle. Now, what exactly are you having trouble with on the second part?

 

[SteamKing]

Okay, I'm still a bit confused on how I would go about doing the second part, but I did manage to get an answer of 39.54% for the third part. Can anyone validate if this is correct or not?

You should know the end points which form the vertices of the triangle and the points of tangency for the sides of the triangle where they touch the circle.

How do you determine that a point is the midpoint of a line segment?

 

[Saracen Rue]

I, too, got 39.54% for the remaining area of the triangle which is not part of the circle. Now, what exactly are you having trouble with on the second part?

The part I'm a bit stuck on is knowing exactly how much I need to prove. Like, with a lot of angles I sort of just made a logical guess at what they were, and it ended up working out. But if I were asked why I made an angle what I did, then I wouldn't really have the proof to explain it. I sort of just assumed the angles would have to be derived from exact values (nothing like 23.34 degrees or something).

You should know the end points which form the vertices of the triangle and the points of tangency for the sides of the triangle where they touch the circle.

How do you determine that a point is the midpoint of a line segment?

Yeah, I get that the midpoint will be half the length of the line segment. To prove that though, you need to express the length of the lines in terms of the radius of the circle. In turn, to do that, you need to be able to prove what certain angles are, and that's the bit I'm stuck on.

 

[SteamKing]

The part I'm a bit stuck on is knowing exactly how much I need to prove. Like, with a lot of angles I sort of just made a logical guess at what they were, and it ended up working out. But if I were asked why I made an angle what I did, then I wouldn't really have the proof to explain it. I sort of just assumed the angles would have to be derived from exact values (nothing like 23.34 degrees or something).

Yeah, I get that the midpoint will be half the length of the line segment. To prove that though, you need to express the length of the lines in terms of the radius of the circle. In turn, to do that, you need to be able to prove what certain angles are, and that's the bit I'm stuck on.

It's not clear what these mysterious angles are that you had to assume their measure.

It's an equilateral triangle you are told to construct. There are exactly 180 degrees in each and every plane triangle, no more, no less. All the angles in an equilateral triangle are equal. There is a right angle between a radius drawn from the center of a circle to a line tangent to the circle. These are simple geometric facts.

You should not have had to guess anything, but since you don't show your work, we don't know what trouble you are having.

If you look at the equilateral triangle circumscribed about the circle, symmetry should also be used to make things easier.