According to http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml it is impossible to construct a triangle from the lengths of its angle bisectors. Is there a more comprehensive account of this problem and its history other than the thesis referenced in the page? Are there any other problems which are similar to this one? (in the sense of showing that something involving angle bisectors is non-constructible)
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Petru Mironescu and Laurentiu Panaitopol, The existence of a triangle with prescribed angle bisector lengths, American Math Monthly 101 (Jan. 1994) 58-60, starts, "Given three arbitrary positive numbers $m,n,p$, does there exist a triangle with angle bisectors of length $m,n,p$? The answer is YES! Moreover, the triangle is unique up to an isometry."
The first paragraph of the second page of the paper deals with the question of (ruler-and-compass) construction, and gives some references.
Added: (Theo Buehler) The passage and the list of references for the convenience of the readers: