Help with a proof - Work Started (Math Club Problem)

Just trying a problem - just seeing if I am on the right track.

**Consider a circle of radius 6.**

- Find A, the area of the inscribed equilateral triangle.
- Find B, the area of the circumscribed equilateral triangle.
- Prove that the geometric mean of A and B is the area of the inscribed hexagon.

I was never taught about the point where the medians of a triangle intersect, the point where the altitudes intersect, and the point where the perpendicular bisectors intersect. I'm pretty sure that information is needed for the problem.

I just don't know where to start. Any help will be appreciated!!

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Re: Help with a proof - Work Started (Math Club Problem)

Didn't you draw a picture? For such problems, drawing the diagram is not only helpful visually, it often can lead to ideas for a proof. Now there are infinitely many equilateral triangles inscribed in a circle of radius 6, same for circumscribed triangles and inscribed hexagons. The areas though are always the same -- needs proof, I guess.

Anyway, here's the "best" figure for proof of your problem:

Attachment 27119

Unless you know some trigonometry, I don't see how you can give the exact areas, but geometrically you can find the relationships among the 3 areas. Good luck.