Two non-congruent triangles have the same area. Each triangle has sides of length and , and the third side of each triangle has integer length. What is the sum of the lengths of the third sides?
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Let the triangles have side lengths and . By Heron's formula, we must have:
Hence, the following polynomial has least two distinct positive integer roots (namely and ):
where is some real number. Note that is even, so its four roots are , , , and . Hence for some and satisfying . We can solve this by inspection and see that and must be and in some order, which yields a final answer of .
Remark: We can verify that the area of a triangle is the same as the area of a triangle, namely .
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