The perimeter of the triangle is 180 units

The perimeter of the triangle is 180 units.

In the above image equilateral triangle ABC is circumscribed about a circle with radius ##=10sqrt3## units. Notice that the sides of the triangle are tangent to the circle, we draw a line segment from the center O to the vertex A, we also draw the radius OD. OA bisects angle A to two 30 degrees parts. And OD is perpendicular to AC. Triangle OAD is 30-60-90 triangle, the sides have a ratio of: ##1 : sqrt3 : 2## , hence: AD ##=10sqrt3*sqrt3=30## units. Since OD bisects AC, we conclude that each side of the triangle ABC is 60 units therefore the perimeter is ##3*60=180## units.