Q:A puzzle board is in the form of an equilateral triangle that has an area of $7\sqrt{3}$ if the board is placed on a circular table, what should be the min area of the table so that the whole board fits inside the table.
A: $\frac{88}{3}$
I get that the side of the triangle is $2\sqrt{7}$ and also that in an equilateral triangle the median, perpendicular bisector, altitude and angle bisector are the same. I'm however still stuck with how to get the radius without resorting to sin/cos etc.
On
In an equilateral triangle the orthocenter, centroid, circumcenter and incenter coincide. The center of the circle is the centroid and height coincides with the median. The radius of the circumcircle is equal to two thirds the height.
Use the property in last sentence to find out the radius of circle that completely circumscribes the triangle using the triangle's height.
Height of triangle = root (21) Radius of circle = 2/3 rd of root (21) Area of circle = 88/3
Hint: Use similar triangles to determine where along the bisectors their intersection lies. Sines and cosines will still be there, of course, since they’re ratios of sides of a right triangle, but you won’t be using the functions explicitly.