Question 1114431
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Let the triangle be ABC, with shorter segment DE and longer segment FG parallel to BC, dividing the triangle into three regions of equal area.<br>
Then triangles ADE, AFG, and ABC are all similar.<br>
Furthermore, the area of triangle ADE is 1/2 the area of triangle AFG; and it is 1/3 the area of triangle ABC.<br>
It is given that the length of FG is 24.  Since the ratio of the areas of triangles ADE and AFG is {{{1:2}}}, the ratio of side lengths between the two triangles is {{{1:sqrt(2)}}}.  So the length of DE is {{{24/sqrt(2) = 12*sqrt(2)}}}.<br>
The ratio of the areas of triangles AFG and ABC is {{{1:3}}}, so the ratio of lengths of sides between those two triangles is {{{1:sqrt(3)}}}.  Since the length of FG is {{{12*sqrt(2)}}}, the length of BC is {{{(12*sqrt(2))*sqrt(3) = 12*sqrt(6)}}}.<br>
Answer: The length of base BC is {{{12*sqrt(6)}}}.