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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?

Solution:
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