Question 1102503
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AO, BO, and CO divide the equilateral triangle into three isosceles triangles, each with an angle measuring 120 degrees.  Each of those triangles can be divided into two 30-60-90 right triangles.<br>
The geometry of 30-60-90 right triangles tells us that the ratio of AO to the side of the triangle is 1:sqrt(3).<br>
Therefore the ratio of AO+BO+CO to the perimeter of the equilateral triangle is 1:sqrt(3).