Thread: 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.

1. Find A, the area of the inscribed equilateral triangle.
2. Find B, the area of the circumscribed equilateral triangle.
3. 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!!

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:



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.