Recently among his posts have been a couple with a related theme, circles inscribed or circumscribed about an equilateral triangle. I'm listing these because they are each a wonderful relationship, and together give these otherwise somewhat mundane seeming triangles a luster students?teachers/others might miss.
I will post the problems, but not the proofs, which (if you can't/won't work them out yourself you can find at the links provided to Antonio's site.
So on we go...
1) draw a circle and inscribe an equilateral triangle. Now pick any point on the circumference and construct segments from this point to the three vertices. The sum of the lengths of the two shorter segments will equal the third. If you want to work on this solution yourself, there is a hint a little below the image, and then a link to the solution as well.
Here is the Hint for the first problem, Think about what you know about the quadrilateral ABDC? The problem, and solution is here.
2) OK:
Same triangle, same circle, but now we sum the square of the three distances ...????? and they sum to twice the square of a side of the equilateral triangle. That proof is here. and the hint???????? it's something you might know about Cosines.
3) And now one with the circle on the inside. Again, from any point on the circle construct segments to the three vertices of the equilateral triangle. Again the sum of the squares is related to a side length, but I'll let you chase that down for yourself. No hint this time, work that out on your own, or you can go to the site here.
Addendum: John Golden sent a comment with a link to a GeoGebra sketch showing all three.
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