Triangle Circumscribed by a Circle (Geometry Problem)

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The discussion revolves around finding the length L of an equilateral triangle that is circumscribed by a circle of radius R. Participants explore geometric relationships, particularly using right triangles formed by bisecting angles and drawing perpendicular lines. One user highlights the connection between a 120-degree angle triangle and the triangle's sides, emphasizing the use of trigonometry to establish the relationship between L and R. Another user acknowledges their lack of formal geometry education, expressing difficulty in visualizing the problem without additional drawings. The conversation concludes with a reminder about the importance of drawing auxiliary lines to clarify geometric relationships.
Saladsamurai
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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?

Picture1-25.png
--->
Picture4-4.png


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.
 
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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.
 
I presume this is what you meant Dick:

Picture5-3.png


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

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