Triangle Circumscribed by a Circle (Geometry Problem)

[Saladsamurai]

Homework Statement

Alright... so I thought of this problem on the train ride home. I am sure plenty thought of it before me...but now it is my turn.

Say want to know what the length L an equilateral triangle must have such that all three points touch the circle of radius R that it is contained by.

I have drawn the problem. i have bisected one of the 60 degree angles to form a right triangle.

I then drew a line perpendicular to the bisecting line to create a similar triangle.

I do not know if this is the way you would have approached this, but it seemed pretty logical to me.

I feel like I have almost all (if not all) the necessary info to write L in terms of R.

Can someone chime in with a hint here?

--->

I feel like I need something more about either the short leg of the inner triangle or its hypoteneuse. then I can relate
both triangles to the angle.

 

[Dick]

You just want the relation between L and R? I see in your picture a triangle with a 120 degree angle at the center between two legs of length R and with an opposite side of length L. It's just trig, saladsamurai, I know you know that. It must have been a rough train ride home.

 

[Saladsamurai]

I presume this is what you meant Dick:

I can honestly say that I may have never seen that if you did not point it out :/

I have never taken a geometry class in my life, so these things do not pop out at me sometimes... arggggghhhh

Thank Dick!
Casey

 

[Dick]

You didn't even need to see it. I also see a 30-60-90 right triangle with a hypotenuse of R and a leg of length L/2. I'm still blaming the train ride home.

 

[Saladsamurai]

Now you're just showing off after I said I suck at Geometry! I STILL do not see that one!

edit: Okay NOW I see it! But I never would have since I still would have needed to have drawn that same line that allowed me
to see the 120 degree triangle.

 

[Dick]

Yeah. You would have gotten it just fine if you'd drawn another radius line. Remember that when you're stuck in the future.