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Find the radius of the circle

Question: ABC is an equilateral triangle with side a. The triangle ABC is circumscribed about a circle with center O. If the length of the side of the equilateral triangle is a, then what is the radius of the circle?

Because ABC is an equilateral triangle, so
angle B = angle C = angle A = 60o
Because AD perpendicular to BC at point D, so triangle BAD is a right triangle.
sin B = AD/AB, so
AD = AB sin B = a sin60o = (square root of 3) a/2
Because BE perpendicular to AC at point E, and ABC is an equilateral triangle, so
BE cuts angle B in half, so angle OBD = 30o
Because OD is a radius and angle OBD = 30o, so
OB = 2r (In a right triangle, the leg opposite 30o is one-half the hypotenuse.)
BE = BO + OE = 2r + r = 3r
Because in an equilateral triangle, the three altitude are equal, so
BE = AD, so 3r = (square root of 3) a/2
so, r = (square root of 3) a/6